What do the motions of a compact disc, a Ferris wheel, a circular saw

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1 RTATIN F RIGID BDIE 9?All segments of a otating helicopte blade have the same angula velocit and angula acceleation. Compaed to a given blade segment, how man times geate is the linea speed of a second segment twice as fa fom the ais of otation? How man times geate is the linea acceleation? What do the motions of a compact disc, a Feis wheel, a cicula saw blade, and a ceiling fan have in common? None of these can be epesented adequatel as a moving point; each involves a bod that otates about an ais that is stationa in some inetial fame of efeence. Rotation occus at all scales, fom the motion of electons in atoms to the motions of entie galaies. We need to develop some geneal methods fo analzing the motion of a otating bod. In this chapte and the net we conside bodies that have definite size and definite shape, and that in geneal can have otational as well as tanslational motion. Real-wold bodies can be ve complicated; the foces that act on them can defom them stetching, twisting, and squeezing them. We ll neglect these defomations fo now and assume that the bod has a pefectl definite and unchanging shape and size. We call this idealized model a igid bod. This chapte and the net ae mostl about otational motion of a igid bod. We begin with kinematic language fo descibing otational motion. Net we look at the kinetic eneg of otation, the ke to using eneg methods fo otational motion. Then in Chapte 0 we ll develop dnamic pinciples that elate the foces on a bod to its otational motion. 9. Angula Velocit and Acceleation In analzing otational motion, let s think fist about a igid bod that otates about a fied ais an ais that is at est in some inetial fame of efeence and does not change diection elative to that fame. The otating igid bod might be a moto shaft, a chunk of beef on a babecue skewe, o a me-go-ound. Figue 9. shows a igid bod (in this case, the indicato needle of a speedomete) otating about a fied ais. The ais passes though point and is LEARNING GAL B studing this chapte, ou will lean: How to descibe the otation of a igid bod in tems of angula coodinate, angula velocit, and angula acceleation. How to analze igid-bod otation when the angula acceleation is constant. How to elate the otation of a igid bod to the linea velocit and linea acceleation of a point on the bod. The meaning of a bod s moment of inetia about a otation ais, and how it elates to otational kinetic eneg. How to calculate the moment of inetia of vaious bodies. 9. A speedomete needle (an eample of a igid bod) otating counteclockwise about a fied ais. The angle u fom the -ais specifies the needle s otational position. Diection of needle s otation u Ais of otation passes though oigin and points out of page. 85

2 86 CHATER 9 Rotation of Rigid Bodies 9. Angula Velocit and Acceleation Measuing angles in adians. (a) ne adian is the angle at which the ac s has the same length as the adius. (b) An angle u in adians is the atio of the ac length s to the adius. ad u 5 s s 5 s 5 u pependicula to the plane of the diagam, which we choose to call the -plane. ne wa to descibe the otation of this bod would be to choose a paticula point on the bod and to keep tack of the - and -coodinates of this point. This isn t a teibl convenient method, since it takes two numbes (the two coodinates and ) to specif the otational position of the bod. Instead, we notice that the line is fied in the bod and otates with it. The angle u that this line makes with the -ais descibes the otational position of the bod; we will use this single quantit u as a coodinate fo otation. The angula coodinate u of a igid bod otating aound a fied ais can be positive o negative. If we choose positive angles to be measued counteclockwise fom the positive -ais, then the angle u in Fig. 9. is positive. If we instead choose the positive otation diection to be clockwise, then u in Fig. 9. is negative. When we consideed the motion of a paticle along a staight line, it was essential to specif the diection of positive displacement along that line; when we discuss otation aound a fied ais, it s just as essential to specif the diection of positive otation. To descibe otational motion, the most natual wa to measue the angle u is not in degees, but in adians. As shown in Fig. 9.a, one adian ( ad) is the angle subtended at the cente of a cicle b an ac with a length equal to the adius of the cicle. In Fig. 9.b an angle u is subtended b an ac of length s on a cicle of adius. The value of u (in adians) is equal to s divided b : u5 s An angle in adians is the atio of two lengths, so it is a pue numbe, without dimensions. If s m and 5.0 m, then u5.5, but we will often wite this as.5 ad to distinguish it fom an angle measued in degees o evolutions. The cicumfeence of a cicle (that is, the ac length all the wa aound the cicle) is p times the adius, so thee ae p (about 6.83) adians in one complete evolution 360. Theefoe ad p imilal, 80 5p ad, 90 5p/ ad, and so on. If we had insisted on measuing the angle u in degees, we would have needed to include an eta facto of p/360 on the ight-hand side of s 5 u in Eq. (9.). B measuing angles in adians, we keep the elationship between angle and distance along an ac as simple as possible. Angula Velocit The coodinate u shown in Fig. 9. specifies the otational position of a igid bod at a given instant. We can descibe the otational motion of such a igid bod in tems of the ate of change of u. We ll do this in an analogous wa to ou desciption of staight-line motion in Chapte. In Fig. 9.3a, a efeence line in a otating bod makes an angle u with the -ais at time t. At a late time t the angle has changed to u. We define the aveage angula velocit v av-z (the Geek lette omega) of the bod in the time inteval Dt 5 t t as the atio of the angula displacement Du 5 u u to Dt: v av-z 5 u u t t o s 5 u 5 Du Dt (9.) (9.) (a) (b) 9.3 (a) Angula displacement Du of a otating bod. (b) Eve pat of a otating (a) Angula displacement igid bod has the same angula velocit Du of the otating needle ove a time inteval Dt: Du/Dt. Du 5 u u Diection of otation at t Du u u The subscipt z indicates that the bod in Fig. 9.3a is otating about the z-ais, which is pependicula to the plane of the diagam. The instantaneous angula velocit v z is the limit of v av-z as Dt appoaches zeo that is, the deivative of u with espect to t: Du v z 5 lim Dt0 Dt 5 du dt (definition of angula velocit) (9.3) When we efe simpl to angula velocit, we mean the instantaneous angula velocit, not the aveage angula velocit. The angula velocit v z can be positive o negative, depending on the diection in which the igid bod is otating (Fig. 9.4). The angula speed v, which we will use etensivel in ections 9.3 and 9.4, is the magnitude of angula velocit. Like odina (linea) speed v, the angula speed is neve negative. CAUTIN Angula velocit vs. linea velocit Keep in mind the distinction between angula velocit v z and odina velocit, o linea velocit, v (see ection.). If an object has a velocit v, the object as a whole is moving along the -ais. B contast, if an object has an angula velocit v z, then it is otating aound the z-ais. We do not mean that the object is moving along the z-ais. Diffeent points on a otating igid bod move diffeent distances in a given time inteval, depending on how fa the point lies fom the otation ais. But because the bod is igid, all points otate though the same angle in the same time (Fig. 9.3b). Hence at an instant, eve pat of a otating igid bod has the same angula velocit. The angula velocit is positive if the bod is otating in the diection of inceasing u and negative if it is otating in the diection of deceasing u. If the angle u is in adians, the unit of angula velocit is the adian pe second (ad/s). the units, such as the evolution pe minute (ev/min o pm), ae often used. ince ev 5 p ad, two useful convesions ae ev/s 5 p ad/s and ev/min 5 pm 5 p 60 ad /s That is, ad/s is about 0 pm. at t 9.4 A igid bod s aveage angula velocit (shown hee) and instantaneous angula velocit can be positive o negative. Counteclockwise otation positive: Du. 0, so v av-z 5 Du/Dt. 0 Du Clockwise otation negative: Du, 0, so v av-z 5 Du/Dt, 0 Du Ais of otation (z-ais) passes though oigin and points out of page.

3 88 CHATER 9 Rotation of Rigid Bodies 9. Angula Velocit and Acceleation 89 Eample 9. Calculating angula velocit The flwheel of a pototpe ca engine is unde test. The angula position u of the flwheel is given b u5.0 ad/s 3 t 3 The diamete of the flwheel is 0.36 m. (a) Find the angle u, in adians and in degees, at times t 5.0 s and t s. (b) Find the distance that a paticle on the im moves duing that time inteval. (c) Find the aveage angula velocit, in ad/s and in ev/min (pm), between t 5.0 s and t s. (d) Find the instantaneous angula velocit at time t 5 t s. LUTIN IDENTIFY: We need to find the values u and u of the angula position at times t and t, the angula displacement Du between t and t, the distance taveled and the aveage angula velocit between t and t, and the instantaneous angula velocit at t. ET U: We e given the angula position u as a function of time, so we can easil find ou fist two taget vaiables u and u ; the angula displacement Du is the diffeence between u and u. Given Du we ll find the distance and the aveage angula velocit using Eqs. (9.) and (9.), espectivel. To find the instantaneous angula velocit, we ll take the deivative of u with espect to time, as in Eq. (9.3). EXECUTE: (a) We substitute the values of t into the given equation: u 5.0 ad/s 3.0 s ad ad p ad 5 90 u 5.0 ad/s s ad ad p ad 5 4, (a) The ight-hand ule fo the diection of the angula velocit vecto v. Revesing the diection of otation eveses the diection of v. (b) The sign of v z fo otation along the z-ais. Angula Velocit As a Vecto As we have seen, ou notation fo the angula velocit v z about the z-ais is eminiscent of the notation v fo the odina velocit along the -ais (see ection.). Just as v is the -component of the velocit vecto v, v z is the z-component of an angula velocit vecto v diected along the ais of otation. As Fig. 9.5a (a) v If ou cul the finges of ou ight hand in the diection of otation ou ight thumb points in the diection of v. (b) The flwheel tuns though an angula displacement of Du 5 u u 5 50 ad 6 ad 5 34 ad. The adius is half the diamete, o 0.8 m. Equation (9.) gives To use Eq. (9.), the angle must be epessed in adians. We dop adians fom the unit fo s because u is eall a dimensionless pue numbe; s is a distance and is measued in metes, the same unit as. (c) In Eq. (9.) we have (d) We use Eq. (9.3): At time t s, v s 5 u m 34 ad 5 4 m v av-z 5 u u t t 5 (b) v points in the positive z-diection: v z. 0 z v 50 ad 6 ad 5.0 s.0 s 5 78 ad/s ad ev 5 78 s p ad 60 s min ev /min v z 5 du dt 5 d dt 3.0 ad /s 3 t ad/s 3 3t ad/s 3 t v z ad/s s 5 50 ad/s EVALUATE: u esult in pat (d) shows that v z is popotional to t and hence inceases with time. u numeical esults ae consistent with this esult: The 50-ad/s instantaneous angula velocit at t s is geate than the 78-ad/s aveage angula velocit fo the 3.0-s inteval leading up to that time (fom t 5.0 s to t s). v points in the negative z-diection: v z, 0 z v shows, the diection of is given b the ight-hand ule that we used to define the vecto poduct in ection.0. If the otation is about the z-ais, then v has onl a z-component; this component is positive if v is along the positive z-ais and negative if v is along the negative z-ais (Fig. 9.5b). The vecto fomulation is especiall useful in situations in which the diection of the otation ais changes. We ll eamine such situations biefl attheend of Chapte 0. In this chapte, howeve, we ll conside onl situations in which the otation ais is fied. Hence thoughout this chapte we ll use angula velocit to efe to v the component of the angula velocit vecto v z, along the ais. v Angula Acceleation When the angula velocit of a igid bod changes, it has an angula acceleation. When ou pedal ou biccle hade to make the wheels tun faste o appl the bakes to bing the wheels to a stop, ou e giving the wheels an angula acceleation. You also impat an angula acceleation wheneve ou change the otation speed of a piece of spinning machine such as an automobile engine s cankshaft. If v z and v z ae the instantaneous angula velocities at times t and t, we define the aveage angula acceleation a av-z ove the inteval Dt 5 t t as the change in angula velocit divided b Dt (Fig. 9.6): The instantaneous angula acceleation is the limit of as Dt 0: Dv z a z 5 lim 5 dv z Dt0 Dt dt a av-z 5 v z v z t t (9.4) (definition of angula acceleation) (9.5) The usual unit of angula acceleation is the adian pe second pe second, o ad/s. Fom now on we will use the tem angula acceleation to mean the instantaneous angula acceleation athe than the aveage angula acceleation. Because v z 5 du/dt, we can also epess angula acceleation as the second deivative of the angula coodinate: a z 5 d dt du dt 5 d u dt a av-z You have pobabl noticed that we ae using Geek lettes fo angula kinematic quantities: u fo angula position, v z fo angula velocit, and a z fo angula acceleation. These ae analogous to fo position, v fo velocit, and a fo acceleation, espectivel, in staight-line motion. In each case, velocit is the ate of change of position with espect to time and acceleation is the ate of change of velocit with espect to time. We will sometimes use the tems linea velocit and linea acceleation fo the familia quantities we defined in Chaptes and 3 to distinguish cleal between these and the angula quantities intoduced in this chapte. In otational motion, if the angula acceleation a z is positive, then the angula velocit v z is inceasing; if a z is negative, then v z is deceasing. The otation is speeding up if a z and v z have the same sign and slowing down if a z and v z have opposite signs. (These ae eactl the same elationships as those between linea acceleation a and linea velocit v fo staight-line motion; see ection.3.) a z 5 Dv z Dt (9.6) 9.6 Calculating the aveage angula acceleation of a otating igid bod. The aveage angula acceleation is the change in angula velocit divided b the time inteval: v z v z v z Dv z t t Dt a av-z 5 5 v z At t At t

4 90 CHATER 9 Rotation of Rigid Bodies 9. Rotation with Constant Angula Acceleation 9 Eample 9. Calculating angula acceleation In Eample 9. we found that the instantaneous angula velocit of the flwheel at an time t is given b v z v z ad/s 3 t (a) Find the aveage angula acceleation between t 5.0 s and t s. (b) Find the instantaneous angula acceleation at time t s. LUTIN IDENTIFY: This eample uses the definitions of aveage angula acceleation and instantaneous angula acceleation a z. a av-z ET U: We ll use Eqs. (9.4) and (9.5) to find the value of between t and t and the value of a z at t 5 t. EXECUTE: (a) The values of v z at the two times ae v z ad/s 3.0 s 5 4 ad/s v z ad/s s 5 50 ad/s a av-z Fom Eq. (9.4) the aveage angula acceleation is (b) Fom Eq. (9.5) the instantaneous angula acceleation at an time t is a z 5 dv z dt At time t s, a av-z 5 50 ad /s 4 ad/s 5.0 s.0 s 5 ad/s 3 t 5 4 ad/s 5 d dt 36.0 ad /s 3 t ad/s 3 t a z 5 ad/s s 5 60 ad/s EVALUATE: Note that the angula acceleation is not constant in this situation. The angula velocit v z is alwas inceasing because a z is alwas positive. Futhemoe, the ate at which the angula velocit inceases is itself inceasing, since inceases with time. a z equal to the aveage value fo an inteval. Using Eq. (9.4) with the inteval fom 0 to t, we find v z 5v 0z a z t a z 5 v z v 0z t 0 (constant angula acceleation onl) (9.7) The poduct a z t is the total change in v z between t 5 0 and the late time t; the angula velocit v z at time t is the sum of the initial value v 0z and this total change. With constant angula acceleation, the angula velocit changes at a unifom ate, so its aveage value between 0 and t is the aveage of the initial and final values: v av-z 5 v 0z v z We also know that v av-z is the total angula displacement uu 0 divided b the time inteval t 0 : o (9.8) 9.7 When the otation ais is fied, the angula acceleation and angula velocit vectos both lie along that ais. a and v in the same diection: Rotation speeding up. a v a and v in the opposite diections: Rotation slowing down. a v Angula Acceleation As a Vecto Just as we did fo angula velocit, it s useful to define an angula acceleation vecto a. Mathematicall, a is the time deivative of the angula velocit vecto v. If the object otates aound the fied z-ais, then a has onl a z-component; the quantit a is just that component. In this case, a is in the same diection as v z if the otation is speeding up and opposite to v if the otation is slowing down (Fig. 9.7). The angula acceleation vecto will be paticulal useful in Chapte 0 when we discuss what happens when the otation ais can change diection. In this chapte, howeve, the otation ais will alwas be fied and we need use onl the z-component a z. When we equate Eqs. (9.8) and (9.9) and multipl the esult b t, we get uu 0 5 v 0z v z t v av-z 5 uu 0 t 0 (9.9) (constant angula acceleation onl) (9.0) To obtain a elationship between u and t that doesn t contain v z, we substitute Eq. (9.7) into Eq. (9.0): uu 0 5 3v 0z v 0z a z t 4t o Test You Undestanding of ection 9. The figue shows a gaph of v z and a z vesus time fo a paticula otating bod. (a) Duing which time intevals is the otation speeding up? (i) 0, t, s; (ii) s, t, 4 s; (iii) 4 s, t, 6 s. (b) Duing which time intevals is the otation slowing down? (i) 0, t, s; (ii) s, t, 4 s; (iii) 4 s, 5, 6 s. 9. Rotation with Constant Angula Acceleation a z vz In Chapte we found that staight-line motion is paticulal simple when the acceleation is constant. This is also tue of otational motion about a fied ais. When the angula acceleation is constant, we can deive equations fo angula velocit and angula position using eactl the same pocedue that we used fo staight-line motion in ection.4. In fact, the equations we ae about to deive ae identical to Eqs. (.8), (.), (.3), and (.4) if we eplace with u, v with v z, and a with a z. We suggest that ou eview ection.4 befoe continuing. Let v 0z be the angula velocit of a igid bod at time t 5 0, and let v z be its angula velocit at an late time t. The angula acceleation is constant and a z t (s) u5u 0 v 0z t a z t (constant angula acceleation onl) (9.) That is, if at the initial time t 5 0 the bod is at angula position u 0 and has angula velocit v 0z, then its angula position u at an late time t is the sum of thee tems: its initial angula position u 0, plus the otation v 0z t it would have if the angula velocit wee constant, plus an additional otation a z t caused b the changing angula velocit. Following the same pocedue as fo staight-line motion in ection.4, we can combine Eqs. (9.7) and (9.) to obtain a elationship between u and v z that does not contain t. We invite ou to wok out the details, following the same pocedue we used to get Eq. (.3). (ee Eecise 9..) In fact, because of the pefect analog between staight-line and otational quantities, we can simpl take Eq. (.3) and eplace each staight-line quantit b its otational analog. We get v z 5v 0z a z uu 0 (constant angula acceleation onl) (9.) CAUTIN Constant angula acceleation Keep in mind that all of these esults ae valid onl when the angula acceleation a z is constant; be caeful not to t to appl them to poblems in which a z is not constant. Table 9. shows the analog between Eqs. (9.7), (9.0), (9.), and (9.) fo fied-ais otation with constant angula acceleation and the coesponding equations fo staight-line motion with constant linea acceleation. NLINE 7.7 Rotational Kinematics

5 9 CHATER 9 Rotation of Rigid Bodies 9.3 Relating Linea and Angula Kinematics 93 Eample A line Q on a otating DVD at t 5 0. Table 9. Compaison of Linea and Angula Motion with Constant Acceleation taight-line Motion with Constant Linea Acceleation Diection of otation Q a 5 constant v 5 v 0 a t 5 0 v 0 t a t v 5 v 0 a v v 0 t Rotation with constant angula acceleation You have just finished watching a movie on DVD and the disc is slowing to a stop. The angula velocit of the disc at t 5 0 is 7.5 ad/s and its angula acceleation is a constant 0.0 ad/s. A line Q on the suface of the disc lies along the -ais at t 5 0 (Fig. 9.8). (a) What is the disc s angula velocit at t s? (b) What angle does the line Q make with the -ais at this time? LUTIN IDENTIFY: The angula acceleation of the disc is constant, so we can use an of the equations deived in this section. u taget vaiables ae the angula velocit and the angula displacement at t s. ET U: We ae given the initial angula velocit v 0z ad/s, the initial angle u between the line Q and the -ais, the angula acceleation a z 50.0 ad/s, and the time t s. Fied-Ais Rotation with Constant Angula Acceleation a z 5 constant v z 5v 0z a z t u5u 0 v 0z t a z t v z 5v 0z a z uu 0 uu 0 5 v z v 0z t With this infomation it s easiest to use Eqs. (9.7) and (9.) to find the taget vaiables v z and u, espectivel. EXECUTE: (a) Fom Eq. (9.7), at t s we have v z 5v 0z a z t ad/s 0.0 ad/s s ad/s (b) Fom Eq. (9.), u5u 0 v 0z t a z t ad/s s 0.0 ad /s s ev ad ad p ad 5.4 ev The DVD has tuned though one complete evolution plus an additional 0.4 evolution that is, though an additional angle of 0.4 ev 360 /ev Hence the line Q is at an angle of 87 with the -ais. EVALUATE: u answe to pat (a) tells us that the angula velocit has deceased. This is as it should be, since a z is negative. We can also use ou answe fo v z in pat (a) to check ou esult fo in pat (b). To do so, we solve Eq. (9.), v u z 5v 0z a z uu 0, fo the angle u: u5u 0 v z v 0z a z ad /s 7.5 ad/s 0.0 ad/s which agees with the esult we found ealie ad Test You Undestanding of ection 9. uppose the DVD in Eample 9.3 was initiall spinning at twice the ate 55.0 ad/s athe than 7.5 ad/s and slowed down at twice the ate 0.0 ad/s athe than 0.0 ad/s. (a) Compaed to the situation in Eample 9.3, how long would it take the DVD to come to a stop? (i) the same amount of time; (ii) twice as much time; (iii) 4 times as much time; (iv) as much time; (v) 4 as much time. (b) Compaed to the situation in Eample 9.3, though how man evolutions would the DVD otate befoe coming to a stop? (i) the same numbe of evolutions; (ii) twice as man evolutions; (iii) 4 times as man evolutions; (iv) as man evolutions; (v) 4 as man evolutions. 9.3 Relating Linea and Angula Kinematics How do we find the linea speed and acceleation of a paticula point in a otating igid bod? We need to answe this question to poceed with ou stud of otation. Fo eample, to find the kinetic eneg of a otating bod, we have to stat fom K 5 mv fo a paticle, and this equies knowing the speed v fo each paticle in the bod. o it s wothwhile to develop geneal elationships between the angula speed and acceleation of a igid bod otating about a fied ais and the linea speed and acceleation of a specific point o paticle in the bod. Linea peed in Rigid-Bod Rotation When a igid bod otates about a fied ais, eve paticle in the bod moves in a cicula path. The cicle lies in a plane pependicula to the ais and is centeed on the ais. The speed of a paticle is diectl popotional to the bod s angula velocit; the faste the bod otates, the geate the speed of each paticle. In Fig. 9.9, point is a constant distance fom the ais of otation, so it moves in a cicle of adius. At an time, the angle u (in adians) and the ac length s ae elated b We take the time deivative of this, noting that is constant fo an specific paticle, and take the absolute value of both sides: Now 0 ds/dt 0 is the absolute value of the ate of change of ac length, which is equal to the instantaneous linea speed v of the paticle. Analogousl, 0 du/dt 0, the absolute value of the ate of change of the angle, is the instantaneous angula speed v that is, the magnitude of the instantaneous angula velocit in ad/s. Thus v 5 v (elationship between linea and angula speeds) (9.3) The fathe a point is fom the ais, the geate its linea speed. The diection of the linea velocit vecto is tangent to its cicula path at each point (Fig. 9.9). CAUTIN peed vs. velocit Keep in mind the distinction between the linea and angula speeds v and v, which appea in Eq. (9.3), and the linea and angula velocities v and v z. The quantities without subscipts, v and v, ae neve negative; the ae the magnitudes of the vectos v and v, espectivel, and thei values tell ou onl how fast a pa- ticle is moving v o how fast a bod is otating v. The coesponding quantities with subscipts, v and v z, can be eithe positive o negative; thei signs tell ou the diection of the motion. Linea Acceleation in Rigid-Bod Rotation We can epesent the acceleation of a paticle moving in a cicle in tems of its centipetal and tangential components, a ad and a tan (Fig. 9.0), as we did in ection 3.4. It would be a good idea to eview that section now. We found that the tangential component of acceleation a tan, the component paallel to the instantaneous velocit, acts to change the magnitude of the paticle s velocit (i.e., the speed) and is equal to the ate of change of speed. Taking the deivative of Eq. (9.3), we find a tan 5 dv dt 5 dv dt 5 a s 5 u ds dt du 5 dt (tangential acceleation of a point on a otating bod) (9.4) 9.9 A igid bod otating about a fied ais though point. v Distance though which point on the bod moves (angle u is in adians) Cicle followed b point Linea speed of point (angula speed v is in ad/s) u v 5 v v s 5 u 9.0 A igid bod whose otation is speeding up. The acceleation of point has a component a ad towad the otation ais (pependicula to v) and a component a tan along the cicle that point follows (paallel to v). Radial and tangential acceleation components: a ad 5 v is point s centipetal acceleation. a tan 5 a means that s otation is speeding up (the bod has angula acceleation). v Linea acceleation of point a tan 5 a a u v 5 v a ad 5 v v s

6 94 CHATER 9 Rotation of Rigid Bodies 9.3 Relating Linea and Angula Kinematics Alwas use adians when elating linea and angula quantities. s 5 u u p/3 ad In an equation that elates linea quantities to angula quantities, the angles MUT be epessed in adians... RIGHT! s 5 (p/3)... neve in degees o evolutions. WRNG s 5 60 Eample 9.4 Thowing a discus This component of a paticle s acceleation is alwas tangent to the cicula path of the paticle. The quantit a5dv/dt in Eq. (9.4) is the ate of change of the angula speed. It is not quite the same as a z 5 dv z/dt, which is the ate of change of the angula velocit. Fo eample, conside a bod otating so that its angula velocit vecto points in the z-diection (Fig. 9.5b). If the bod is gaining angula speed at a ate of 0 ad/s pe second, then a50 ad/s. But v z is negative and becoming moe negative as the otation gains speed, so a z 50 ad/s. The ule fo otation about a fied ais is that a is equal to a z if v z is positive but equal to a z if v z is negative. The component of the paticle s acceleation diected towad the otation ais, the centipetal component of acceleation a ad, is associated with the? change of diection of the paticle s velocit. In ection 3.4 we woked out the elationship a ad 5 v /. We can epess this in tems of v b using Eq. (9.3): a ad 5 v 5v (centipetal acceleation of a point on a otating bod) (9.5) This is tue at each instant, even when v and v ae not constant. The centipetal component alwas points towad the ais of otation. The vecto sum of the centipetal and tangential components of acceleation of a paticle in a otating bod is the linea acceleation a (Fig. 9.0). CAUTIN Use angles in adians in all equations It s impotant to emembe that Eq. (9.), s 5 u, is valid onl when u is measued in adians. The same is tue of an equation deived fom this, including Eqs. (9.3), (9.4), and (9.5). When ou use these equations, ou must epess the angula quantities in adians, not evolutions o degees (Fig. 9.). Equations (9.), (9.3), and (9.4) also appl to an paticle that has the same tangential velocit as a point in a otating igid bod. Fo eample, when a ope wound aound a cicula clinde unwaps without stetching o slipping, its speed and acceleation at an instant ae equal to the speed and tangential acceleation of the point at which it is tangent to the clinde. The same pinciple holds fo situations such as biccle chains and spockets, belts and pulles that tun without slipping, and so on. We will have seveal oppotunities to use these elationships late in this chapte and in Chapte 0. Note that Eq. (9.5) fo the centipetal component a ad is applicable to the ope o chain onl at points that ae in contact with the clinde o spocket. the points do not have the same acceleation towad the cente of the cicle that points on the clinde o spocket have. 9. (a) Whiling a discus in a cicle. (b) u sketch showing the acceleation components fo the discus. (a) a tan EVALUATE: Note that we dopped the unit adian fom ou esults fo a tan, a ad, and a. We can do this because adian is a dimensionless quantit. The magnitude a is about nine times g, the acceleation due to gavit. Can ou show that if the angula speed doubles to Eample 9.5 (a) a ad Designing a popelle You ae asked to design an aiplane popelle to tun at 400 pm. The fowad aispeed of the plane is to be 75.0 m/s 70 km/h, o about 68 mi/h, and the speed of the tips of the popelle blades though the ai must not eceed 70 m/s (Fig. 9.3a). (This is about 0.80 times the speed of sound in ai. If the popelle tips wee to move too close to the speed of sound, the would poduce a temendous amount of noise.) (a) What is the maimum adius the popelle can have? (b) With this adius, what is the acceleation of the popelle tip? LUTIN IDENTIFY: The object of inteest in this eample is a paticle at the tip of the popelle; ou taget vaiables ae the paticle s distance fom the ais and its acceleation. Note that the speed of this paticle though the ai (which cannot eceed 70 m/s) is due to both the popelle s otation and the fowad motion of the aiplane. a (b) 0.0 ad/s while a emains the same, the acceleation magnitude a inceases to 3 m/s, o almost 33g? 9.3 (a) A popelle-diven aiplane in flight. (b) u sketch showing the velocit components fo the popelle tip. (b) vtip ET U: As Fig. 9.3b shows, the velocit of a paticle at the popelle tip is the vecto sum of its tangential velocit due to the popelle s otation (magnitude v tan, given b Eq. (9.3)) and the fowad velocit of the aiplane (magnitude v plane m/s). The otation plane of the popelle is pependicula to the diection of flight, so these two vectos ae pependicula and we can use the thagoean theoem to elate v tan and v plane to v tip. We will then set v tip 5 70 m/s and solve fo the adius. Note that the angula speed of the popelle is constant, so the acceleation of the popelle tip has onl a adial component; we ll find it using Eq. (9.5). EXECUTE: We fist convet v to ad/s (see Fig. 9.): ev v5400 pm min 5 5 ad/s p ad ev min 60 s A discus thowe moves the discus in a cicle of adius 80.0 cm. At a cetain instant, the thowe is spinning at an angula speed of 0.0 ad/s and the angula speed is inceasing at 50.0 ad/s. At this instant, find the tangential and centipetal components of the acceleation of the discus and the magnitude of the acceleation. LUTIN IDENTIFY: We model the discus as a paticle taveling on a cicula path (Fig. 9.a), so we can use the ideas developed in this section. eation components a tan and a ad, which we ll find with Eqs. (9.4) and (9.5), espectivel. Given these components of the acceleation vecto, we ll find its magnitude a (the thid taget vaiable) using the thagoean theoem. EXECUTE: Fom Eqs. (9.4) and (9.5), a tan 5 a m 50.0 ad/s m/s a ad 5v ad/s m m/s v tan 5 v v plane m/s 400 ev/min ET U: We ae given the adius m, the angula speed v50.0 ad/s, and the ate of change of angula speed a ad/s. (Fig. 9.b).The fist two taget vaiables ae the accel- The magnitude of the acceleation vecto is a 5 "a tan a ad m/s Continued

7 96 CHATER 9 Rotation of Rigid Bodies 9.4 Eneg in Rotational Motion 97 (a) Fom Fig. 9.3b and Eq. (9.3), the velocit magnitude is given b v tip 5 v plane v tan 5 v plane v so 5 v tip v plane and 5 "v tip v plane v v If v tip 5 70 m/s, the popelle adius is 5 " 70 m /s 75.0 m/s 5 ad/s 5.03 m v total (b) The centipetal acceleation is a ad 5v 5 5 ad/s.03 m m/s The tangential acceleation is zeo because the angula speed is constant. EVALUATE: Fom gf 5 ma, the popelle must eet a foce of N on each kilogam of mateial at its tip! This is wh popelles ae made out of tough mateial, usuall aluminum allo. i same plane, so we specif that is the pependicula distance fom the ais to the ith paticle. When a igid bod otates about a fied ais, the speed v i of the ith paticle is given b Eq. (9.3), v i 5 i v, whee v is the bod s angula speed. Diffeent paticles have diffeent values of, but v is the same fo all (othewise, the bod wouldn t be igid). The kinetic eneg of the ith paticle can be epessed as m i v i 5 m i i v The total kinetic eneg of the bod is the sum of the kinetic enegies of all its paticles: Conceptual Eample 9.6 Biccle geas K 5 m v m v c 5 ai m i i v How ae the angula speeds of the two biccle spockets in Fig. 9.4 elated to the numbe of teeth on each spocket? 9.4 The spockets and chain of a biccle. Taking the common facto v / out of this epession, we get LUTIN The chain does not slip o stetch, so it moves at the same tangential speed v on both spockets. Fom Eq. (9.3), v 5 font v font 5 ea v ea so v ea v font 5 font ea v ea ea Rea spocket v font K 5 m m c v 5 a i m i i v The quantit in paentheses, obtained b multipling the mass of each paticle b the squae of its distance fom the ais of otation and adding these poducts, is denoted b I and is called the moment of inetia of the bod fo this otation ais: The angula speed is invesel popotional to the adius. This elationship also holds fo pulles connected b a belt, povided the belt doesn t slip. Fo chain spockets the teeth must be equall spaced on the cicumfeences of both spockets fo the chain to mesh popel with both. Let N font and N ea be the numbes of teeth; the condition that the tooth spacing is the same on both spockets is p font N font 5 p ea N ea Combining this with the othe equation, we get o v ea v font 5 N font N ea font ea 5 N font N ea v v font Font spocket The angula speed of each spocket is invesel popotional to the numbe of teeth. n a multispeed bike, ou get the highest angula speed v ea of the ea wheel fo a given pedaling ate v font when the atio N font/n ea is maimum; this means using the lagestadius font spocket (lagest N font ) and the smallest-adius ea spocket (smallest N ea ). I 5 m m c 5 ai m i i (definition of moment of inetia) (9.6) The wod moment means that I depends on how the bod s mass is distibuted in space; it has nothing to do with a moment of time. Fo a bod with a given otation ais and a given total mass, the geate the distance fom the ais to the paticles that make up the bod, the geate the moment of inetia. In a igid bod, the distances i ae all constant and I is independent of how the bod otates aound the given ais. The I unit of moment of inetia is the kilogam-mete kg # m. In tems of moment of inetia I, the otational kinetic eneg K of a igid bod is 9.5 An appaatus fee to otate aound a vetical ais. To va the moment of inetia, the two equal-mass clindes can be locked into diffeent positions on the hoizontal shaft. Mass close to ais mall moment of inetia Eas to stat appaatus otating NLINE 7.7 Rotational Inetia Test You Undestanding of ection 9.3 Infomation is stoed on a CD o DVD (see Fig. 9.8) in a coded patten of tin pits. The pits ae aanged in a tack that spials outwad towad the im of the disc. As the disc spins inside a plae, the tack is scanned at a constant linea speed. How must the otation speed of the disc change as the plae s scanning head moves ove the tack? (i) The otation speed must incease. (ii) The otation speed must decease. (iii) The otation speed must sta the same. 9.4 Eneg in Rotational Motion A otating igid bod consists of mass in motion, so it has kinetic eneg. As we will see, we can epess this kinetic eneg in tems of the bod s angula speed and a new quantit, called moment of inetia, that depends on the bod s mass and how the mass is distibuted. To begin, we think of a bod as being made up of a lage numbe of paticles, with masses m, m, c at distances,, c fom the ais of otation. We label the paticles with the inde i: The mass of the ith paticle is m i and its distance fom the ais of otation is i. The paticles don t necessail all lie in the K 5 Iv (otational kinetic eneg of a igid bod) (9.7) The kinetic eneg given b Eq. (9.7) is not a new fom of eneg; it s simpl the sum of the kinetic enegies of the individual paticles that make up the otating igid bod. To use Eq. (9.7), v must be measued in adians pe second, not evolutions o degees pe second, to give K in joules. That s because we used v i 5 i v in ou deivation. Equation (9.7) gives a simple phsical intepetation of moment of inetia: The geate the moment of inetia, the geate the kinetic eneg of a igid bod otating with a given angula speed v. We leaned in Chapte 6 that the kinetic eneg of a bod equals the amount of wok done to acceleate that bod fom est. o the geate a bod s moment of inetia, the hade it is to stat the bod otating if it s at est and the hade it is to stop its otation if it s alead otating (Fig. 9.5). Fo this eason, I is also called the otational inetia. The net eample shows how changing the otation ais can affect the value of I. Rotation ais Mass fathe fom ais Geate moment of inetia Hade to stat appaatus otating Rotation ais

8 98 CHATER 9 Rotation of Rigid Bodies 9.4 Eneg in Rotational Motion 99 Eample 9.7 Moments of inetia fo diffeent otation aes An enginee is designing a machine pat consisting of thee heav disks linked b lightweight stuts (Fig. 9.6). (a) What is the moment of inetia of this bod about an ais though the cente of disk A, pependicula to the plane of the diagam? (b) What is the moment of inetia about an ais though the centes of disks B and C? (c) If the bod otates about an ais though A pependicula to the plane of the diagam, with angula speed v54.0 ad/s, what is its kinetic eneg? LUTIN IDENTIFY: We ll conside the disks as massive paticles and the lightweight stuts as massless ods. Then we can use the ideas of 9.6 An oddl shaped machine pat. Ais passing though disks B and C B m B kg this section to calculate the moment of inetia of this collection of thee paticles. ET U: In pats (a) and (b), we ll use Eq. (9.6) to find the moments of inetia fo each of the two aes. Given the moment of inetia fo ais A, we ll use Eq. (9.7) in pat (c) to find the otational kinetic eneg. EXECUTE: (a) The paticle at point A lies on the ais. Its distance fom the ais is zeo, so it contibutes nothing to the moment of inetia. Equation (9.6) gives I 5 a m i i kg 0.50 m 0.0 kg 0.40 m kg # m (b) The paticles at B and C both lie on the ais, so fo them 5 0 and neithe contibutes to the moment of inetia. nl A contibutes, and we have I 5 a m i i kg 0.40 m kg # m Table 9. Moments of Inetia of Vaious Bodies (a) lende od, ais though cente I 5 ML L (b) lende od, ais though one end I 5 ML 3 L (c) Rectangula plate, ais though cente a I 5 Ma b (e) Hollow clinde (f) olid clinde (g) Thin-walled hollow clinde I 5 MR R I 5 MR I 5 MR b (d) Thin ectangula plate, ais along edge a (h) olid sphee I 5 MR 5 I 5 Ma 3 b (i) Thin-walled hollow sphee I 5 MR 3 Ais passing though disk A 0.50 m 0.30 m (c) Fom Eq. (9.7), K 5 Iv kg # m 4.0 ad / s J A m A kg 0.40 m C m C kg EVALUATE: u esults show that the moment of inetia fo the ais though A is geate than that fo the ais though B and C. Hence, of the two aes, it s easie to make the machine pat otate about the ais though B and C. R R R R R R CAUTIN Moment of inetia depends on the choice of ais The esults of pats (a) and (b) of Eample 9.7 show that the moment of inetia of a bod depends on the location and oientation of the ais. It s not enough to just sa, The moment of inetia of this bod is kg # m. We have to be specific and sa, The moment of inetia of this bod about the ais though B and C is kg # m. In Eample 9.7 we epesented the bod as seveal point masses, and we evaluated the sum in Eq. (9.6) diectl. When the bod is a continuous distibution of matte, such as a solid clinde o plate, the sum becomes an integal, and we need to use calculus to calculate the moment of inetia. We will give seveal eamples of such calculations in ection 9.6; meanwhile, Table 9. gives moments of inetia fo seveal familia shapes in tems of thei masses and dimensions. Each bod shown in Table 9. is unifom; that is, the densit has the same value at all points within the solid pats of the bod. CAUTIN Computing the moment of inetia You ma be tempted to t to compute the moment of inetia of a bod b assuming that all the mass is concentated at the cente of mass and multipling the total mass b the squae of the distance fom the cente of mass to the ais. Resist that temptation; it doesn t wok! Fo eample, when a unifom thin od of length L and mass M is pivoted about an ais though one end, pependicula to the od, the moment of inetia is I 5 ML /3 [case (b) in Table 9.]. If we took the mass as concentated at the cente, a distance L/ fom the ais, we would obtain the incoect esult I 5 M L/ 5 ML /4. Now that we know how to calculate the kinetic eneg of a otating igid bod, we can appl the eneg pinciples of Chapte 7 to otational motion. Hee ae some points of stateg and some eamples. oblem-olving tateg 9. Rotational Eneg IDENTIFY the elevant concepts: You can use wok eneg elationships and consevation of eneg to find elationships involving position and motion of a igid bod otating aound a fied ais. As we saw in Chapte 7, the eneg method is usuall not helpful fo poblems that involve elapsed time. In Chapte 0 we ll see how to appoach otational poblems of this kind. ET U the poblem using the same steps as in oblem-olving tateg 7. (ection 7.), with the following addition: 5. Man poblems involve a ope o cable wapped aound a otating igid bod, which functions as a pulle. In these situations, emembe that the point on the pulle that contacts the ope has the same linea speed as the ope, povided the ope doesn t slip on the pulle. You can then take advantage of Eqs. (9.3) and (9.4), which elate the linea speed and tangential acceleation of a point on a igid bod to the angula velocit and angula acceleation of the bod. Eamples 9.8 and 9.9 illustate this point. EXECUTE the solution: As in Chapte 7, wite epessions fo the initial and final kinetic and potential enegies K, K, U, and U and the nonconsevative wok W othe (if an). The new featue is otational kinetic eneg, which is epessed in tems of the bod s moment of inetia I fo the given ais and its angula speed v AK 5 Iv B instead of its mass m and speed v. ubstitute these epessions into K U W othe 5 K U (if nonconsevative wok is done) o K U 5 K U (if onl consevative wok is done) and solve fo the taget vaiable(s). As in Chapte 7, it s helpful to daw ba gaphs showing the initial and final values of K, U, and E 5 K U. EVALUATE ou answe: As alwas, check whethe ou answe makes phsical sense.

9 300 CHATER 9 Rotation of Rigid Bodies 9.5 aallel-ais Theoem 30 Eample A cable unwinds fom a clinde (side view). 9.0 N Eample 9.9 An unwinding cable I A light, fleible, nonstetching cable is wapped seveal times aound a winch dum, a solid clinde of mass 50 kg and diamete 0.0 m, which otates about a stationa hoizontal ais held b fictionless beaings (Fig. 9.7). The fee end of the cable is pulled with a constant 9.0-N foce fo a distance of.0 m. It unwinds without slipping and tuns the clinde. If the clinde is initiall at est, find its final angula speed and the final speed of the cable. LUTIN IDENTIFY: We will solve this poblem using eneg methods. oint is when the clinde fist begins to move, and point is when the cable has moved.0 m. We ll assume that the light cable is massless, so that onl the clinde has kinetic eneg. The clinde doesn t move veticall, so thee ae no changes in gavitational potential eneg. Thee is fiction between the cable and the clinde, which is what makes the clinde otate when the cable is pulled. But because the cable doesn t slip, thee is no sliding of the cable elative to the clinde and no mechanical eneg is lost in fiction. Because the cable is massless, the foce that the cable eets on the clinde im is equal to the applied foce F..0 m 50 kg An unwinding cable II 0.0 m We wap a light, fleible cable aound a solid clinde with mass M and adius R. The clinde otates with negligible fiction about a stationa hoizontal ais. We tie the fee end of the cable to a block of mass m and elease the object with no initial velocit at a distance h above the floo. As the block falls, the cable unwinds without stetching o slipping, tuning the clinde. Find the speed of the falling block and the angula speed of the clinde just as the block stikes the floo. LUTIN IDENTIFY: As in Eample 9.8, the cable doesn t slip and fiction does no wok. The cable does no net wok; at its uppe end the foce and displacement ae in the same diection, and at its lowe end the ae in opposite diections. Thus the total wok done b the two ends of the cable is zeo. Hence onl gavit does wok, and so mechanical eneg is conseved. ET U: Figue 9.8a shows the situation just befoe the block begins to fall. At this point the sstem has no kinetic eneg, so ET U: The clinde stats at est, so the initial kinetic eneg is K 5 0. Between points and the foce F does wok on the clinde ove a distance s 5.0 m. As a esult, the kinetic eneg at point is K 5 Iv. ne of ou taget vaiables is v; the othe is the speed of the cable at point, which is equal to the tangential speed v of the clinde at that point. We ll find v fom v b using Eq. (9.3). EXECUTE: The wok done on the clinde is W othe 5 Fs N.0 m 5 8 J. Fom Table 9. the moment of inetia is 9.8 u sketches fo this poblem. (a) I 5 mr 5 50 kg m kg # m (The adius R is half the diamete of the clinde.) The elationship K U W othe 5 K U then gives 0 0 W othe 5 Iv 0 W othe v5 Å I 5 0 ad/s (b) 5 Å 8 J kg # m The final tangential speed of the clinde, and hence the final speed of the cable, is v 5 Rv m 0 ad/s 5. m/s EVALUATE: If the mass of the cable can t be neglected, then some of the wok done would go into the kinetic eneg of the cable. Hence the clinde would end up with less kinetic eneg and a smalle angula speed than we calculated hee. K 5 0. We take the potential eneg to be zeo when the block is EXECUTE: We use ou epessions fo K, U, K, and U and the at floo level; then U 5 mgh and U 5 0. (We can ignoe the gavitational potential eneg fo the otating clinde, since its height U 5 K U. We then solve fo v: elationship v5v/r in the eneg-consevation equation K doesn t change.) Just befoe the block hits the floo (Fig. 9.8b), both the block and the clinde have kinetic eneg. The total 0 mgh 5 kinetic eneg K at that instant is mv v MR R 0 5 m M v gh K 5 mv v 5 Å Iv M/m The final angula speed of the clinde is v5v/r. Fom Table 9. the moment of inetia of the clinde is I 5 MR. EVALUATE: Let s check some paticula cases. When M is much Also, v and v ae elated b v 5 Rv, since the speed of the falling lage than m, v is ve small, as we would epect. When M is block must be equal to the tangential speed at the oute suface of much smalle than m, v is neal equal to!gh, which is the the clinde. We ll use these elationships to solve fo the taget speed of a bod that falls feel fom an initial height h. Does it vaiables v and v shown in Fig. 9.8b. supise ou that v doesn t depend on the adius of the clinde? Gavitational otential Eneg fo an Etended Bod In Eample 9.9 the cable was of negligible mass, so we could ignoe its kinetic eneg as well as the gavitational potential eneg associated with it. If the mass is not negligible, we need to know how to calculate the gavitational potential eneg associated with such an etended bod. If the acceleation of gavit g is the same at all points on the bod, the gavitational potential eneg is the same as though all the mass wee concentated at the cente of mass of the bod. uppose we take the -ais veticall upwad. Then fo a bod with total mass M, the gavitational potential eneg U is simpl U 5 Mg cm (gavitational potential eneg fo an etended bod) (9.8) whee cm is the -coodinate of the cente of mass. This epession applies to an etended bod, whethe it is igid o not (Fig. 9.9). To pove Eq. (9.8), we again epesent the bod as a collection of mass elements m i. The potential eneg fo element is m i g i, so the total potential eneg is m i U 5 m g m g c 5 m m c g But fom Eq. (8.8), which defines the coodinates of the cente of mass, m m c 5 m m c cm 5 M cm whee M 5 m m c is the total mass. Combining this with the above epession fo U, we find U 5 Mg cm in ageement with Eq. (9.8). We leave the application of Eq. (9.8) to the poblems. We ll make use of this elationship in Chapte 0 in the analsis of igid-bod poblems in which the ais of otation moves. Test You Undestanding of ection 9.4 uppose the clinde and block in Eample 9.9 have the same mass, so m 5 M. Just befoe the block stikes the floo, which statement is coect about the elationship between the kinetic eneg of the falling block and the otational kinetic eneg of the clinde? (i) The block has moe kinetic eneg than the clinde. (ii) The block has less kinetic eneg than the clinde. (iii) The block and the clinde have equal amounts of kinetic eneg. 9.5 aallel-ais Theoem We pointed out in ection 9.4 that a bod doesn t have just one moment of inetia. In fact, it has infinitel man, because thee ae infinitel man aes about which it might otate. But thee is a simple elationship between the moment of inetia of a bod of mass M about an ais though its cente of I cm NLINE 7. Woman and Flwheel Elevato Eneg Appoach 7.3 Rotoide Eneg Appoach 9.9 In a technique called the Fosbu flop afte its innovato, this athlete aches his bod as he passes ove the ba in the high jump. As a esult, his cente of mass actuall passes unde the ba. This technique equies a smalle incease in gavitational potential eneg [Eq. (9.8)] than the olde method of staddling the ba. cm

10 30 CHATER 9 Rotation of Rigid Bodies 9.6 Moment-of-Inetia Calculations The mass element m i has coodinates i, i with espect to an ais of otation though the cente of mass (cm) and coodinates i a, i b with espect to the paallel ais though point. Ais of otation passing though cm and pependicula to the plane of the figue i a d i b b Mass element m i i a m i cm i econd ais of otation paallel to the one though the cm I mass and the moment of inetia about an othe ais paallel to the oiginal one but displaced fom it b a distance d. This elationship, called the paallelais theoem, states that I 5 I cm Md (paallel-ais theoem) (9.9) To pove this theoem, we conside two aes, both paallel to the z-ais, one though the cente of mass and the othe though a point (Fig. 9.0). Fist we take a ve thin slice of the bod, paallel to the -plane and pependicula to the z-ais. We take the oigin of ou coodinate sstem to be at the cente of mass of the bod; the coodinates of the cente of mass ae then cm 5 cm 5 z cm 5 0. The ais though the cente of mass passes though this thin slice at point, and the paallel ais passes though point, whose - and -coodinates ae a, b. The distance of this ais fom the ais though the cente of mass is d, whee d 5 a b. We can wite an epession fo the moment of inetia I about the ais though point. Let m i be a mass element in ou slice, with coodinates i, i, z i. Then the moment of inetia I cm of the slice about the ais though the cente of mass (at ) is I cm 5 ai m i i i Test You Undestanding of ection 9.5 A pool cue is a wooden od with a unifom composition and tapeed with a lage diamete at one end than at the othe end. Use the paallel-ais theoem to decide whethe a pool cue has a lage moment of inetia (i) fo an ais though the thicke end of the od and pependicula to the length of the od, o (ii) fo an ais though the thinne end of the od and pependicula to the length of the od. *9.6 Moment-of-Inetia Calculations NTE: This optional section is fo students who ae familia with integal calculus. If a igid bod is a continuous distibution of mass like a solid clinde o a solid sphee it cannot be epesented b a few point masses. In this case the sum of masses and distances that defines the moment of inetia [Eq. (9.6)] becomes an integal. Imagine dividing the bod into elements of mass dm that ae ve small, so that all points in a paticula element ae at essentiall the same pependicula distance fom the ais of otation. We call this distance, as befoe. Then the moment of inetia is Eample 9.0 lice of a bod of mass M Using the paallel-ais theoem A pat of a mechanical linkage (Fig. 9.) has a mass of 3.6 kg. We measue its moment of inetia about an ais 0.5 m fom its cente of mass to be I kg # m. What is the moment of inetia I cm about a paallel ais though the cente of mass? 9. Calculating I cm fom a measuement of I. cm 0.5 m Ais though cente of mass Ais though The moment of inetia of the slice about the ais though is I 5 ai m i 3 i a i b 4 These epessions don t involve the coodinates z i measued pependicula to the slices, so we can etend the sums to include all paticles in all slices. Then I becomes the moment of inetia of the entie bod fo an ais though. We then epand the squaed tems and egoup, and obtain I 5 ai m i i i a ai m i i b ai m i i a b ai m i The fist sum is I cm. Fom Eq. (8.8), the definition of the cente of mass, the second and thid sums ae popotional to cm and cm ; these ae zeo because we have taken ou oigin to be the cente of mass. The final tem is d multiplied b the total mass, o Md. This completes ou poof that I 5 I cm Md. As Eq. (9.9) shows, a igid bod has a lowe moment of inetia about an ais though its cente of mass than about an othe paallel ais. Thus it s easie to stat a bod otating if the otation ais passes though the cente of mass. This suggests that it s somehow most natual fo a otating bod to otate about an ais though its cente of mass; we ll make this idea moe quantitative in Chapte 0. LUTIN IDENTIFY: The paallel-ais theoem allows us to elate the moments of inetia I cm and I though the two paallel aes. ET U: We ll use Eq. (9.9) to detemine the taget vaiable I cm. EXECUTE: Reaanging the equation and substituting the values, I cm 5 I Md kg # m 3.6 kg 0.5 m kg # m EVALUATE: u esult shows that I cm is less than I. This is as it should be: As we saw ealie, the moment of inetia fo an ais though the cente of mass is lowe than fo an othe paallel ais. I 5 3 dm (9.0) To evaluate the integal, we have to epesent and dm in tems of the same integation vaiable. When the object is effectivel one-dimensional, such as the slende ods (a) and (b) in Table 9., we can use a coodinate along the length and elate dm to an incement d. Fo a thee-dimensional object it is usuall easiest to epess dm in tems of an element of volume dv and the densit of the bod. Densit is mass pe unit volume, 5dm/dV, so we ma also wite Eq. (9.0) as I 5 3 dv This epession tells us that a bod s moment of inetia depends on how its densit vaies within its volume (Fig. 9.). If the bod is unifom in densit, then we ma take outside the integal: I 5 3 dv (9.) To use this equation, we have to epess the volume element dv in tems of the diffeentials of the integation vaiables, such as dv 5 d d dz. The element dv must alwas be chosen so that all points within it ae at ve neal the same distance fom the ais of otation. The limits on the integal ae detemined b the shape and dimensions of the bod. Fo egulal shaped bodies, this integation is often eas to do. Eample 9. Unifom thin od, ais pependicula to length Figue 9.3 shows a slende unifom od with mass M and length L. It might be a baton held b a twile in a maching band (less the ubbe end caps). Compute its moment of inetia about an ais though, at an abita distance h fom one end. LUTIN IDENTIFY: The od is a continuous distibution of mass, so we must use integation to find the moment of inetia. We choose as an element of mass a shot section of od with length d at a distance fom point. M 9. B measuing small vaiations in the obits of satellites, geophsicists can measue the eath s moment of inetia. This tells us how ou planet s mass is distibuted within its inteio. The data show that the eath is fa dense at the coe than in its oute laes. 9.3 Finding the moment of inetia of a thin od about an ais though. Ais L h L h Mass element: od segment of length d d

11 304 CHATER 9 Rotation of Rigid Bodies 9.6 Moment-of-Inetia Calculations 305 ET U: The atio of the mass dm of an element to the total mass M is equal to the atio of its length d to the total length L: We ll detemine I fom Eq. (9.0) with eplaced b (see Fig. 9.3). EXECUTE: Figue 9.3 shows that the integation limits on ae fom h to L h. Hence we obtain Eample 9. dm M 5 d L 5 M L 3 3 T Lh Hollow o solid clinde, otating about ais of smmet Figue 9.4 shows a hollow, unifom clinde with length L, inne adius R, and oute adius R. It might be a steel clinde in a pinting pess o a sheet-steel olling mill. Find the moment of inetia about the ais of smmet of the clinde. LUTIN IDENTIFY: Again we must use integation to find the moment of inetia, but now we choose as a volume element a thin clindical shell of adius, thickness d, and length L. All pats of this element ae at ve neal the same distance fom the ais. ET U: The volume of the element is ve neal equal to that of a flat sheet with thickness d, length L, and width p (the cicumfeence of the shell). Then We will use this epession in Eq. (9.0) and integate fom 5 R to 5 R. EXECUTE: The moment of inetia is given b 5 pl R R R R It is usuall moe convenient to epess the moment of inetia in tems of the total mass M of the bod, which is its densit multiplied b the total volume V. The volume is so the total mass M is I 5 3 dm 5 M Lh L 3 d h dm 5 dv 5 pl d I 5 3 dm 5 3 R 5 pl 3 R 5 pl 4 so dm 5 M L d h 5 3 M L 3Lh 3h R 3 d R pl d R 4 R 4 V 5pL R R M 5V 5pL R R EVALUATE: Fom this geneal epession we can find the moment of inetia about an ais though an point on the od. Fo eample, if the ais is at the left end, h 5 0 and If the ais is at the ight end, we should get the same esult. utting h 5 L, we again get If the ais passes though the cente, the usual place fo a twiled baton, then h 5 L/ and These esults agee with the epessions in Table Finding the moment of inetia of a hollow clinde about its smmet ais. Mass element: clindical shell with adius and thickness d Hence the moment of inetia is I 5 3 ML I 5 3 ML I 5 ML I 5 M R R EVALUATE: This agees with Table 9., case (e). If the clinde is solid, such as a lawn olle, R 5 0. Calling the oute adius R simpl R, we find that the moment of inetia of a solid clinde of adius R is I 5 MR If the clinde has a ve thin wall (like a pipe), R and R ae ve neal equal; if R epesents this common adius, I 5 MR Ais R R We could have pedicted this last esult; in a thin-walled clinde, all the mass is the same distance 5 R fom the ais, so I 5 dm 5 R dm 5 MR. d L Eample 9.3 Unifom sphee with adius R, ais though cente Find the moment of inetia of a solid, unifom sphee (like a billiad ball o ball beaing) about an ais though its cente. LUTIN IDENTIFY: To calculate the moment of inetia we divide the sphee into thin disks of thickness d (Fig. 9.5), whose moment of inetia we know fom Eample 9.. We ll integate ove these to find the total moment of inetia. The onl tick point is that the adius and mass of a disk depend on its distance fom the cente of the sphee. ET U: The adius of the disk shown in Fig. 9.5 is Its volume is 5 "R dv 5p d 5p R d 9.5 Finding the moment of inetia of a sphee about an ais though its cente. d R Mass element: disk of adius and thickness d located a distance fom the cente of the sphee and its mass is dm 5 dv 5p R d EXECUTE: Fom Eample 9., the moment of inetia of a disk of adius and mass dm is di 5 dm 5 A"R B 3p R d4 5 p R d Integating this epession fom 5 0 to 5 R gives the moment of inetia of the ight hemisphee. The total I fo the entie sphee, including both hemisphees, is just twice this: R p I 5 3 R d Caing out the integation, we obtain Test You Undestanding of ection 9.6 Two hollow clindes have the same inne and oute adii and the same mass, but the have diffeent lengths. ne is made of low-densit wood and the othe of high-densit lead. Which clinde has the geate moment of inetia aound its ais of smmet? (i) the wood clinde; (ii) the lead clinde; (iii) the two moments of inetia ae equal. Ais I 5 8p 5 R5 The mass M of the sphee of volume V 5 4pR 3 /3 is M 5V 5 4pR3 3 B compaing the epessions fo I and M, we find 0 I 5 5 MR EVALUATE: This esult agees with the epession in Table 9., case (h). Note that the moment of inetia of a solid sphee of mass M and adius R is less than the moment of inetia of a solid clinde of the same mass and adius, I 5 MR. The eason is that moe of the sphee s mass is located close to the ais.

12 CHATER 9 UMMARY Discussion Questions 307 Ke Tems Rotational kinematics: When a igid bod otates about a stationa ais (usuall called the z-ais), its position is descibed b an angula coodinate u. The angula velocit v z is the time deivative of u, and the angula acceleation a z is the time deivative of v z o the second deivative of u. (ee Eamples 9. and 9..) If the angula acceleation is constant, then u, v z, and a z ae elated b simple kinematic equations analogous to those fo staight-line motion with constant linea acceleation. (ee Eample 9.3.) Relating linea and angula kinematics: The angula speed v of a igid bod is the magnitude of its angula velocit. The ate of change of v is a5dv/dt. Fo a paticle in the bod a distance fom the otation ais, the speed v and the components of the acceleation a ae elated to v and a. (ee Eamples ) Moment of inetia and otational kinetic eneg: The moment of inetia I of a bod about a given ais is a measue of its otational inetia: The geate the value of I, the moe difficult it is to change the state of the bod s otation. The moment of inetia can be epessed as a sum ove the paticles m i that make up the bod, each of which is at its own pependicula distance i fom the ais. The otational kinetic eneg of a igid bod otating about a fied ais depends on the angula speed v and the moment of inetia I fo that otation ais. (ee Eamples ) Calculating the moment of inetia: The paallel-ais theoem elates the moments of inetia of a igid bod of mass M about two paallel aes: an ais though the cente of mass (moment of inetia I cm ) and a paallel ais a distance d fom the fist ais (moment of inetia I ). (ee Eample 9.0.) If the bod has a continuous mass distibution, the moment of inetia can be calculated b integation. (ee Eamples ) Du v z 5 lim Dt0 Dt 5 du dt Dv z a z 5 lim 5 dv z 5 d u Dt0 Dt dt dt u5u 0 v 0z t a z t (constant (constant onl) uu 0 5 v z v 0z t v z 5v 0z a z t (constant onl) a z a z a z onl) v z 5v 0z a z uu 0 (constant onl) v 5 v a z a tan 5 dv dt 5 dv dt 5 a a ad 5 v 5v I 5 m m c 5 ai m i i K 5 Iv I 5 I cm Md (9.3) (9.5) (9.) (9.0) (9.7) (9.) (9.3) (9.4) (9.5) (9.6) (9.7) (9.9) v z 5 du dt Linea acceleation of point Ais of otation m Mass M At t At t v Icm Du u u v 3 cm a I 5 I cm Md a z 5 dv z dt a tan 5 a v 5 v a ad 5 v s u m I 5 m i i i K 5 Iv m 3 d igid bod, 85 adian, 86 aveage angula velocit, 86 angula displacement, 86 instantaneous angula velocit, 87 Answe to Chapte pening Question? Both segments of the igid blade have the same angula speed v. Fom Eqs. (9.3) and (9.5), doubling the distance fo the same v doubles the linea speed v 5 v and doubles the adial acceleation a ad 5v. Answes to Test You Undestanding Questions aveage angula acceleation, 89 instantaneous angula acceleation, 89 angula speed, 93 tangential component of acceleation, 93 centipetal component of acceleation, Answes: (a) (i) and (iii), (b) (ii) The otation is speeding up when the angula velocit and angula acceleation have the same sign, and slowing down when the have opposite signs. Hence it is speeding up fo 0, t, s (v z and a z ae both positive) and fo 4 s, t, 6 s (v z and a z ae both negative), but is slowing down fo s, t, 4 s (v z is positive and a z is negative). Note that the bod is otating in one diection fo t, 4 s (v z is positive) and in the opposite diection fo t. 4 s (v z is negative). 9. Answes: (a) (i), (b) (ii) When the DVD comes to est, v z 5 0. Fom Eq. (9.7), the time when this occus is t 5 v z v 0z /a z 5 v 0z/a z (this is a positive time because a z is negative). If we double the initial angula velocit v 0z and also double the angula acceleation a z, thei atio is unchanged and the otation stops in the same amount of time. The angle though which the DVD otates is given b Eq. (9.0): uu 0 5 v 0z v z t 5 v 0z t RBLEM Discussion Questions Q9.. Which of the following fomulas is valid if the angula acceleation of an object is not constant? Eplain ou easoning in each case. (a) v 5 v; (b) a (c) (d) a tan 5 v tan 5 a; v5v 0 at; ; (e) K 5 Iv. Q9.. A diatomic molecule can be modeled as two point masses, m and m, slightl sepaated (Fig. 9.6). If the molecule is oiented along the -ais, it has kinetic eneg K when it spins about Figue 9.6 Question Q9.. m z m moment of inetia, 97 otational kinetic eneg, 97 paallel-ais theoem, 30 (since the final angula velocit is v z 5 0). The initial angula velocit v 0z has been doubled but the time t is the same, so the angula displacement uu 0 (and hence the numbe of evolutions) has doubled. You can also come to the same conclusion using Eq. (9.). 9.3 Answe: (ii) Fom Eq. (9.3), v 5 v. To maintain a constant linea speed v, the angula speed v must decease as the scanning head moves outwad (geate ). 9.4 Answe: (i) The kinetic eneg in the falling block is mv, and the kinetic eneg in the otating clinde is A mr BARB v 5 Iv 5 4 mv. Hence the total kinetic eneg of the sstem 3 is 4 mv, of which two-thids is in the block and one-thid is in the clinde. 9.5 Answe: (ii) Moe of the mass of the pool cue is concentated at the thicke end, so the cente of mass is close to that end. The moment of inetia though a point at eithe end is I 5 I cm Md ; the thinne end is fathe fom the cente of mass, so the distance d and the moment of inetia I ae geate fo the thinne end. 9.6 Answe: (iii) u esult fom Eample 9. does not depend on the clinde length L. The moment of inetia depends onl on the adial distibution of mass, not on its distibution along the ais. Fo instucto-assigned homewok, go to the -ais. What will its kinetic eneg (in tems of K) be if it spins at the same angula speed about (a) the z-ais and (b) the -ais? Q9.3. What is the diffeence between tangential and adial acceleation fo a point on a otating bod? Q9.4. In Fig. 9.4, all points on the chain have the same linea speed. Is the magnitude of the linea acceleation also the same fo all points on the chain? How ae the angula acceleations of the two spockets elated? Eplain. Q9.5. In Fig. 9.4, how ae the adial acceleations of points at the teeth of the two spockets elated? Eplain the easoning behind ou answe. Q9.6. A flwheel otates with constant angula velocit. Does a point on its im have a tangential acceleation? A adial acceleation? Ae these acceleations constant in magnitude? In diection? In each case give the easoning behind ou answe. Q9.7. What is the pupose of the spin ccle of a washing machine? Eplain in tems of acceleation components. Q9.8. Although angula velocit and angula acceleation can be teated as vectos, the angula displacement u, despite having a magnitude and a diection, cannot. This is because u does not follow the commutative law of vecto addition (Eq..3). ove this to ouself in the following wa: La ou phsics tetbook flat on 306

13 308 CHATER 9 Rotation of Rigid Bodies Eecises 309 the desk in font of ou with the cove side up so ou can ead the witing on it. Rotate it though 90 about a hoizontal ais so that the fathest edge comes towad ou. Call this angula displacement u. Then otate it b 90 about a vetical ais so that the left edge comes towad ou. Call this angula displacement u. The spine of the book should now face ou, with the witing on it oiented so that ou can ead it. Now stat ove again but ca out the two otations in the evese ode. Do ou get a diffeent esult? That is, does u u equal u u? Now epeat this epeiment but this time with an angle of athe than 90. Do ou think that the infinitesimal displacement du obes the commutative law of addition and hence qualifies as a vecto? If so, how is the diection of du elated to the diection of v? Q9.9. Can ou think of a bod that has the same moment of inetia fo all possible aes? If so, give an eample, and if not, eplain wh this is not possible. Can ou think of a bod that has the same moment of inetia fo all aes passing though a cetain point? If so, give an eample and indicate whee the point is located. Q9.0. To maimize the moment of inetia of a flwheel while minimizing its weight, what shape and distibution of mass should it have? Eplain. Q9.. How might ou detemine epeimentall the moment of inetia of an iegulal shaped bod about a given ais? Q9.. A clindical bod has mass M and adius R. Can the mass be distibuted within the bod in such a wa that its moment of inetia about its ais of smmet is geate than MR? Eplain. Q9.3. Descibe how ou could use pat (b) of Table 9. to deive the esult in pat (d). Q9.4. A hollow spheical shell of adius R that is otating about an ais though its cente has otational kinetic eneg K. If ou want to modif this sphee so that it has thee times as much kinetic eneg at the same angula speed while keeping the same mass, what should be its adius in tems of R? Q9.5. Fo the equations fo I given in pats (a) and (b) of Table 9. to be valid, must the od have a cicula coss section? Is thee an estiction on the size of the coss section fo these equations to appl? Eplain. Q9.6. In pat (d) of Table 9., the thickness of the plate must be much less than a fo the epession given fo I to appl. But in pat (c), the epession given fo I applies no matte how thick the plate is. Eplain. Q9.7. Two identical balls, A and B, ae each attached to ve light sting, and each sting is wapped aound the im of a fictionless pulle of mass M. The onl diffeence is that the pulle fo ball A is a solid disk, while the one fo ball B is a hollow disk, like pat (e) in Table 9.. If both balls ae eleased fom est and fall the same distance, which one will have moe kinetic eneg, o will the have the same kinetic eneg? Eplain ou easoning. Q9.8. An elaboate pulle consists of fou identical balls at the ends of spokes etending out fom a otating dum (Fig. 9.7). A bo is connected to a light thin ope wound aound the im of the dum. When it is eleased fom est, the bo acquies a speed V afte having fallen a distance d. Now the fou balls ae moved inwad close to the dum, and the bo is again eleased fom est. Afte it has fallen a distance d, will its speed be equal to V, geate than V, o less than V? how o eplain wh. Figue 9.7 Question 9.8. Dum Bo Q9.9. You can use an angula measue adians, degees, o evolutions in some of the equations in Chapte 9, but ou can use onl adian measue in othes. Identif those fo which using adians is necessa and those fo which it is not, and in each case give the easoning behind ou answe. Q9.0. When calculating the moment of inetia of an object, can we teat all its mass as if it wee concentated at the cente of mass of the object? Justif ou answe. Q9.. A wheel is otating about an ais pependicula to the plane of the wheel and passing though the cente of the wheel. The angula speed of the wheel is inceasing at a constant ate. oint A is on the im of the wheel and point B is midwa between the im and cente of the wheel. Fo each of the following quantities, is its magnitude lage at point A, at point B, o is it the same at both points? (a) angula speed; (b) tangential speed; (c) angula acceleation; (d) tangential acceleation; (e) adial acceleation. Justif each of ou answes. Eecises ection 9. Angula Velocit and Acceleation 9.. (a) What angle in adians is subtended b an ac.50 m long on the cicumfeence of a cicle of adius.50 m? What is this angle in degees? (b) An ac 4.0 cm long on the cicumfeence of a cicle subtends an angle of 8. What is the adius of the cicle? (c) The angle between two adii of a cicle with adius.50 m is ad. What length of ac is intecepted on the cicumfeence of the cicle b the two adii? 9.. An aiplane popelle is otating at 900 pm ev/min. (a) Compute the popelle s angula velocit in ad/s. (b) How man seconds does it take fo the popelle to tun though 35? 9.3. The angula velocit of a flwheel obes the equation v z t 5 A Bt, whee t is in seconds and A and B ae constants having numeical values.75 (fo A) and.50 (fo B). (a) What ae the units of A and B if v is in ad/s? (b) What is the angula acceleation of the wheel at (i) t and (ii) t s? (c) Though what angle does the flwheel tun duing the fist.00 s? (Hint: ee ection.6.) 9.4. A fan blade otates with angula velocit given b v z t 5 gbt, whee g55.00 ad/s and b ad/s 3. (a) Calculate the angula acceleation as a function of time. (b) Calculate the instantaneous angula acceleation a z at t s and the aveage angula acceleation a av-z fo the time inteval t 5 0 to t s. How do these two quantities compae? If the ae diffeent, wh ae the diffeent? 9.5. A child is pushing a me-go-ound. The angle though which the me-go-ound has tuned vaies with time accoding to u t 5gt bt 3, whee g ad/s and b50.00 ad/s 3. (a) Calculate the angula velocit of the me-go-ound as a function of time. (b) What is the initial value of the angula velocit? (c) Calculate the instantaneous value of the angula velocit v z at t s and the aveage angula velocit v av-z fo the time inteval t 5 0 to t s. how that v av-z is not equal to the aveage of the instantaneous angula velocities at t 5 0 and t s, and eplain wh it is not At t 5 0 the cuent to a dc electic moto is evesed, esulting in an angula displacement of the moto shaft given b u t 5 50 ad/s t 0.0 ad/s t.50 ad/s 3 t 3. (a) At what time is the angula velocit of the moto shaft zeo? (b) Calculate the angula acceleation at the instant that the moto shaft has zeo angula velocit. (c) How man evolutions does the moto shaft tun though between the time when the cuent is evesed and the instant when the angula velocit is zeo? (d) How fast was the moto shaft otating at t 5 0, when the cuent was evesed? (e) Calculate the aveage angula velocit fo the time peiod fom t 5 0 to the time calculated in pat (a) The angle u though which a disk dive tuns is given b u t 5 a bt ct 3, whee a, b, and c ae constants, t is in seconds, and u is in adians. When t 5 0, u5p/4 ad and the angula velocit is.00 ad/s, and when t 5.50 s, the angula acceleation is.5 ad/s. (a) Find a, b, and c, including thei units. (b) What is the angula acceleation when u5p/4 ad? (c) What ae u and the angula velocit when the angula acceleation is 3.50 ad/s? 9.8. A wheel is otating about an ais that is in the z-diection. The angula velocit v z is 6.00 ad/s at t 5 0, inceases lineal with time, and is 8.00 m/s at t s. We have taken counteclockwise otation to be positive. (a) Is the angula acceleation duing this time inteval positive o negative? (b) Duing what time inteval is the speed of the wheel inceasing? Deceasing? (c) What is the angula displacement of the wheel at t s? ection 9. Rotation with Constant Angula Acceleation 9.9. A biccle wheel has an initial angula velocit of.50 ad/s. (a) If its angula acceleation is constant and equal to ad/s, what is its angula velocit at t 5.50 s? (b) Though what angle has the wheel tuned between t 5 0 and t 5.50 s? 9.0. An electic fan is tuned off, and its angula velocit deceases unifoml fom 500 ev/min to 00 ev/min in 4.00 s. (a) Find the angula acceleation in ev/s and the numbe of evolutions made b the moto in the 4.00-s inteval. (b) How man moe seconds ae equied fo the fan to come to est if the angula acceleation emains constant at the value calculated in pat (a)? 9.. The otating blade of a blende tuns with constant angula acceleation.50 ad/s. (a) How much time does it take to each an angula velocit of 36.0 ad/s, stating fom est? (b) Though how man evolutions does the blade tun in this time inteval? 9.. (a) Deive Eq. (9.) b combining Eqs. (9.7) and (9.) to eliminate t. (b) The angula velocit of an aiplane popelle inceases fom.0 ad/s to 6.0 ad/s while tuning though 7.00 ad. What is the angula acceleation in ad/s? 9.3. A tuntable otates with a constant.5 ad/s angula acceleation. Afte 4.00 s it has otated though an angle of 60.0 ad. What was the angula velocit of the wheel at the beginning of the 4.00-s inteval? 9.4. A cicula saw blade 0.00 m in diamete stats fom est. In 6.00 s it acceleates with constant angula acceleation to an angula velocit of 40 ad/s. Find the angula acceleation and the angle though which the blade has tuned A high-speed flwheel in a moto is spinning at 500 pm when a powe failue suddenl occus. The flwheel has mass 40.0 kg and diamete 75.0 cm. The powe is off fo 30.0 s, and duing this time the flwheel slows due to fiction in its ale beaings. Duing the time the powe is off, the flwheel makes 00 complete evolutions. (a) At what ate is the flwheel spinning when the powe comes back on? (b) How long afte the beginning of the powe failue would it have taken the flwheel to stop if the powe had not come back on, and how man evolutions would the wheel have made duing this time? 9.6. A compute disk dive is tuned on stating fom est and has constant angula acceleation. If it took s fo the dive to make its second complete evolution, (a) how long did it take to make the fist complete evolution, and (b) what is its angula acceleation, in ad/s? 9.7. A safet device bings the blade of a powe mowe fom an initial angula speed of v to est in.00 evolution. At the same constant acceleation, how man evolutions would it take the blade to come to est fom an initial angula speed v 3 that was thee times as geat, v 3 5 3v? 9.8. A staight piece of eflecting tape etends fom the cente of a wheel to its im. You daken the oom and use a camea and stobe unit that flashes once eve s to take pictues of the wheel as it otates counteclockwise. You tigge the stobe so that the fist flash t 5 0 occus when the tape is hoizontal to the ight at an angula displacement of zeo. Fo the following situations daw a sketch of the photo ou will get fo the time eposue ove five flashes (at t 5 0, s, 0.00 s, 0.50 s, and 0.00 s), and gaph u vesus t and v vesus t fo t 5 0 to t s. (a) The angula velocit is constant at 0.0 ev/s. (b) The wheel stats fom est with a constant angula acceleation of 5.0 ev/s. (c) The wheel is otating at 0.0 ev/s at t 5 0 and changes angula velocit at a constant ate of 50.0 ev/s At t 5 0 a ginding wheel has an angula velocit of 4.0 ad/s. It has a constant angula acceleation of 30.0 ad/s until a cicuit beake tips at t 5.00 s. Fom then on, it tuns though 43 ad as it coasts to a stop at constant angula acceleation. (a) Though what total angle did the wheel tun between t 5 0 and the time it stopped? (b) At what time did it stop? (c) What was its acceleation as it slowed down? ection 9.3 Relating Linea and Angula Kinematics 9.0. In a chaming 9thcentu hotel, an old-stle ele- Figue 9.8 Eecise 9.0. vato is connected to a counteweight b a cable that passes ove a otating disk.50 m in diamete (Fig. 9.8). The elevato is aised and loweed b Disk tuning the disk, and the cable does not slip on the im of the disk but tuns with it. (a) At how man pm must the disk tun to aise the elevato at 5.0 cm/s? (b) To stat the elevato moving, it must be acceleated at 8 g. What must be the angula acceleation of the disk, in ad/s? (c) Though what angle (in adians and degees) has the disk Elevato Counteweight tuned when it has aised the elevato 3.5 m between floos? 9.. Using astonomical data fom Appendi F, along with the fact that the eath spins on its ais once pe da, calculate (a) the eath s obital angula speed (in ad/s) due to its motion aound the sun, (b) its angula speed (in ad/s) due to its aial spin, (c) the tangential speed of the eath aound the sun (assuming a cicula obit), (d) the tangential speed of a point on the eath s equato due to the planet s aial spin, and (e) the adial and tangential acceleation components of the point in pat (d). 9.. Compact Disc. A compact disc (CD) stoes music in a coded patten of tin pits 0 7 m deep. The pits ae aanged in a tack that spials outwad towad the im of the disc; the inne and oute adii of this spial ae 5.0 mm and 58.0 mm, espectivel. As the disc spins inside a CD plae, the tack is scanned at a constant linea speed of.5 m/s. (a) What is the angula speed of the CD when the innemost pat of the tack is scanned? The

14 30 CHATER 9 Rotation of Rigid Bodies Eecises 3 outemost pat of the tack? (b) The maimum plaing time of a CD is 74.0 min. What would be the length of the tack on such a maimum-duation CD if it wee stetched out in a staight line? (c) What is the aveage angula acceleation of a maimum-duation CD duing its 74.0-min plaing time? Take the diection of otation of the disc to be positive A wheel of diamete 40.0 cm stats fom est and otates with a constant angula acceleation of 3.00 ad/s. At the instant the wheel has computed its second evolution, compute the adial acceleation of a point on the im in two was: (a) using the elationship a and (b) fom the elationship a ad 5 v ad 5v / Ultacentifuge. Find the equied angula speed (in ev/min of an ultacentifuge fo the adial acceleation of a point.50 cm fom the ais to equal 400,000g (that is, 400,000 times the acceleation due to gavit) A flwheel with a adius of m stats fom est and acceleates with a constant angula acceleation of ad/s. Compute the magnitude of the tangential acceleation, the adial acceleation, and the esultant acceleation of a point on its im (a) at the stat; (b) afte it has tuned though 60.0 ; (c) afte it has tuned though An electic tuntable m in diamete is otating about a fied ais with an initial angula velocit of 0.50 ev/s and a constant angula acceleation of ev/s. (a) Compute the angula velocit of the tuntable afte 0.00 s. (b) Though how man evolutions has the tuntable spun in this time inteval? (c) What is the tangential speed of a point on the im of the tuntable at t s? (d) What is the magnitude of the esultant acceleation of a point on the im at t s? 9.7. Centifuge. An advetisement claims that a centifuge takes up onl 0.7 m of bench space but can poduce a adial acceleation of 3000g at 5000 ev/min. Calculate the equied adius of the centifuge. Is the claim ealistic? 9.8. (a) Deive an equation fo the adial acceleation that includes v and v, but not. (b) You ae designing a me-go-ound fo which a point on the im will have a adial acceleation of m/s when the tangential velocit of that point has magnitude.00 m/s. What angula velocit is equied to achieve these values? 9.9. Electic Dill. Accoding to the shop manual, when dilling a.7-mm-diamete hole in wood, plastic, o aluminum, a dill should have a speed of 50 ev/min. Fo a.7-mm-diamete dill bit tuning at a constant 50 ev/min, find (a) the maimum linea speed of an pat of the bit and (b) the maimum adial acceleation of an pat of the bit At t s a point on the im of a 0.00-m-adius wheel has a tangential speed of 50.0 m/s as the wheel slows down with a tangential acceleation of constant magnitude 0.0 m/s. (a) Calculate the wheel s constant angula acceleation. (b) Calculate the angula velocities at t s and t 5 0. (c) Though what angle did the wheel tun between t 5 0 and t s? (d) At what time will the adial acceleation equal g? 9.3. The spin ccles of a washing machine have two angula speeds, 43 ev/min and 640 ev/min. The intenal diamete of the dum is m. (a) What is the atio of the maimum adial foce on the laund fo the highe angula speed to that fo the lowe speed? (b) What is the atio of the maimum tangential speed of the laund fo the highe angula speed to that fo the lowe speed? (c) Find the laund s maimum tangential speed and the maimum adial acceleation, in tems of g You ae to design a otating clindical ale to lift 800-N buckets of cement fom the gound to a ooftop 78.0 m above the gound. The buckets will be attached to a hook on the fee end of a cable that waps aound the im of the ale; as the ale tuns, the buckets will ise. (a) What should the diamete of the ale be in ode to aise the buckets at a stead.00 cm/s when it is tuning at 7.5 pm? (b) If instead the ale must give the buckets an upwad acceleation of m/s, what should the angula acceleation of the ale be? While iding a multispeed biccle, the ide can select the adius of the ea spocket that is fied to the ea ale. The font spocket of a biccle has adius.0 cm. If the angula speed of the font spocket is ev/s, what is the adius of the ea spocket fo which the tangential speed of a point on the im of the ea wheel will be 5.00 m/s? The ea wheel has adius m. ection 9.4 Eneg in Rotational Motion Fou small sphees, each of Figue 9.9 Eecise which ou can egad as a point of m mass 0.00 kg, ae aanged in a squae m on a side and 0.00 kg connectedbetemellightods A B (Fig. 9.9). Find the moment of inetia of the sstem about an ais (a) though the cente of the squae, pependicula to its plane (an ais though point in the figue); (b) bisecting two opposite sides of the squae (an ais along the line AB in the figue); (c) that passes though the centes of the uppe left and lowe ight sphees and though point Calculate the moment of inetia of each of the following unifom objects about the aes indicated. Consult Table 9. as needed. (a) A thin.50-kg od of length 75.0 cm, about an ais pependicula to it and passing though (i) one end and (ii) its cente, and (iii) about an ais paallel to the od and passing though it. (b) A 3.00-kg sphee 38.0 cm in diamete, about an ais though its cente, if the sphee is (i) solid and (ii) a thin-walled hollow shell. (c) An 8.00-kg clinde, of length 9.5 cm and diamete.0 cm, about the cental ais of the clinde, if the clinde is (i) thinwalled and hollow, and (ii) solid mall blocks, each with mass m, ae clamped at the ends and at the cente of a od of length L and negligible mass. Compute the moment of inetia of the sstem about an ais pependicula to the od and passing though (a) the cente of the od and (b) a point one-fouth of the length fom one end A unifom ba has two small balls glued to its ends. The ba is.00 m long and has mass 4.00 kg, while the balls each have mass kg and can be teated as point masses. Find the moment of inetia of this combination about each of the following aes: (a) an ais pependicula to the ba though its cente; (b) an ais pependicula to the ba though one of the balls; (c) an ais paallel to the ba though both balls; (d) an ais paallel to the ba and m fom it A twile s baton is made of a slende metal clinde of mass M and length L. Each end has a ubbe cap of mass m, and ou can accuatel teat each cap as a paticle in this poblem. Find the total moment of inetia of the baton about the usual twiling ais (pependicula to the baton though its cente) A wagon wheel is constucted as shown in Fig The adius of the wheel is m, and the im has mass.40 kg. Each of the eight spokes that lie along a diamete and Figue 9.30 Eecise m ae m long has mass 0.80 kg. What is the moment of inetia of the wheel about an ais though its cente and pependicula to the plane of the wheel? (Use the fomulas given in Table 9..) A unifom disk of adius R is cut in half so that the emaining half has mass M (Fig. 9.3a). (a) What is the moment of inetia of this half about an ais pependicula to its plane though point A? (b) Wh did ou answe in pat (a) come out the same as if this wee a complete disk of mass M? (c) What would be the moment of inetia of a quate disk of mass M and adius R about an ais pependicula to its plane passing though point B (Fig. 9.3b)? 9.4. A compound disk of outside diamete 40.0 cm is made up of a unifom solid disk of adius 50.0 cm and aea densit 3.00 g/cm suounded b a concentic ing of inne adius 50.0 cm, oute adius 70.0 cm, and aea densit.00 g/cm. Find the moment of inetia of this object about an ais pependicula to the plane of the object and passing though its cente An aiplane popelle is.08 m in length (fom tip to tip) with mass 7 kg and Figue 9.3 Eecise is otating at 400 pm ev/min about an R ais though its cente. You can model the popelle as a slende od. (a) What is its otational kinetic eneg? (b) uppose that, due to weight constaints, ou had to educe the popelle s mass to 75.0% of its oiginal mass, but ou still needed to keep the same size and kinetic eneg. What would its angula speed have to be, in pm? Eneg fom the Moon? uppose that some time in the futue we decide to tap the moon s otational eneg fo use on eath. In additional to the astonomical data in Appendi F, ou ma need to know that the moon spins on its ais once eve 7.3 das. Assume that the moon is unifom thoughout. (a) How much total eneg could we get fom the moon s otation? (b) The wold pesentl uses about J of eneg pe ea. If in the futue the wold uses five times as much eneg eal, fo how man eas would the moon s otation povide us eneg? In light of ou answe, does this seem like a cost-effective eneg souce in which to invest? You need to design an industial tuntable that is 60.0 cm in diamete and has a kinetic eneg of 0.50 J when tuning at 45.0 pm ev/min. (a) What must be the moment of inetia of the tuntable about the otation ais? (b) If ou wokshop makes this tuntable in the shape of a unifom solid disk, what must be its mass? The flwheel of a gasoline engine is equied to give up 500 J of kinetic eneg while its angula velocit deceases fom 650 ev/min to 50 ev/min. What moment of inetia is equied? A light, fleible ope is wapped seveal times aound a hollow clinde, with a weight of 40.0 N and a adius of 0.5 m, that otates without fiction about a fied hoizontal ais. The clinde is attached to the ale b spokes of a negligible moment of inetia. The clinde is initiall at est. The fee end of the ope is pulled with a constant foce fo a distance of 5.00 m, at which point the end of the ope is moving at 6.00 m/s. If the ope does not slip on the clinde, what is the value of? Eneg is to be stoed in a 70.0-kg flwheel in the shape of a unifom solid disk with adius R 5.0 m. To pevent stuctual failue of the flwheel, the maimum allowed adial acceleation of a point on its im is 3500 m/s. What is the maimum kinetic eneg that can be stoed in the flwheel? (a) A (b) B R M M uppose the solid clinde in the appaatus descibed in Eample 9.9 (ection 9.4) is eplaced b a thin-walled, hollow clinde with the same mass M and adius R. The clinde is attached to the ale b spokes of a negligible moment of inetia. (a) Find the speed of the hanging mass m just as it stikes the floo. (b) Use eneg concepts to eplain wh the answe to pat (a) is diffeent fom the speed found in Eample A fictionless pulle has the shape of a unifom solid disk of mass.50 kg and adius 0.0 cm. A.50-kg stone is attached to a ve light wie that is wapped aound the im of the pulle (Fig. 9.3), and the sstem is eleased fom est. (a) How fa must the stone fall so that the pulle has 4.50 J of kinetic eneg? (b) What pecent of the total kinetic eneg does the pulle have? A bucket of mass m is tied Figue 9.3 Eecise kg stone.50-kg pulle to a massless cable that is wapped aound the oute im of a fictionless unifom pulle of adius R, simila to the sstem shown in Fig In tems of the stated vaiables, what must be the moment of inetia of the pulle so that it alwas has half as much kinetic eneg as the bucket? 9.5. How I cales. If we multipl all the design dimensions of an object b a scaling facto f, its volume and mass will be multiplied b f 3. (a) B what facto will its moment of inetia be multiplied? (b) If a 48-scale model has a otational kinetic eneg of.5 J, what will be the kinetic eneg fo the full-scale object of the same mateial otating at the same angula velocit? 9.5. A unifom.00-m ladde of mass 9.00 kg is leaning against a vetical wall while making an angle of 53.0 with the floo. A woke pushes the ladde up against the wall until it is vetical. How much wok did this peson do against gavit? A unifom 3.00-kg ope 4.0 m long lies on the gound at the top of a vetical cliff. A mountain climbe at the top lets down half of it to help his patne climb up the cliff. What was the change in potential eneg of the ope duing this maneuve? ection 9.5 aallel-ais Theoem Find the moment of inetia of a hoop (a thin-walled, hollow ing) with mass M and adius R about an ais pependicula to the hoop s plane at an edge About what ais will a unifom, balsa-wood sphee have the same moment of inetia as does a thin-walled, hollow, lead sphee of the same mass and adius, with the ais along a diamete? Use the paallel-ais theoem to show that the moments of inetia given in pats (a) and (b) of Table 9. ae consistent A thin, ectangula sheet of metal has mass M and sides of length a and b. Use the paallel-ais theoem to calculate the moment of inetia of the sheet fo an ais that is pependicula to the plane of the sheet and that passes though one cone of the sheet (a) Fo the thin ectangula plate shown in pat (d) of Table 9., find the moment of inetia about an ais that lies in the plane of the plate, passes though the cente of the plate, and is paallel to the ais shown in the figue. (b) Find the moment of inetia of the plate fo an ais that lies in the plane of the plate, passes though the cente of the plate, and is pependicula to the ais in pat (a).

15 3 CHATER 9 Rotation of Rigid Bodies oblems A thin unifom od of mass M and length L is bent at its cente so that the two segments ae now pependicula to each othe. Find its moment of inetia about an ais pependicula to its plane and passing though (a) the point whee the two segments meet and (b) the midpoint of the line connecting its two ends. *ection 9.6 Moment-of-Inetia Calculations *9.60. Using the infomation in Table 9. and the paallel-ais theoem, find the moment of inetia of the slende od with mass M and length L shown in Fig. 9.3 about an ais though, at an abita distance h fom one end. Compae ou esult to that found b integation in Eample 9. (ection 9.6). *9.6. Use Eq. (9.0) to calculate the moment of inetia of a unifom, solid disk with mass M and adius R fo an ais pependicula to the plane of the disk and passing though its cente. *9.6. Use Eq. (9.0) to calculate the moment of inetia of a slende, unifom od with mass M and length L about an ais at one end, pependicula to the od. *9.63. A slende od with length L has a mass pe unit length that vaies with distance fom the left end, whee 5 0, accoding to dm/d 5g, whee g has units of kg/m. (a) Calculate the total mass of the od in tems of g and L. (b) Use Eq. (9.0) to calculate the moment of inetia of the od fo an ais at the left end, pependicula to the od. Use the epession ou deived in pat (a) to epess I in tems of M and L. How does ou esult compae to that fo a unifom od? Eplain this compaison. (c) Repeat pat (b) fo an ais at the ight end of the od. How do the esults fo pats (b) and (c) compae? Eplain this esult. oblems ketch a wheel ling in the plane of ou pape and otating counteclockwise. Choose a point on the im and daw a vecto fom the cente of the wheel to that point. (a) What is the diection of v? (b) how that the velocit of the point is v 5v 3. (c) how that the adial acceleation of the point is a v 3 v 5v 3 v 3 ad 5 (see Eecise 9.8) Tip to Mas. You ae woking on a poject with NAA to launch a ocket to Mas, with the ocket blasting off fom eath when eath and Mas ae aligned along a staight line fom the sun. If Mas is now 60 ahead of eath in its obit aound the sun, when should ou launch the ocket? (Note: All the planets obit the sun in the same diection, ea on Mas is.9 eath-eas, and assume cicula obits fo both planets.) A olle in a pinting pess tuns though an angle u t given b u t 5gt bt 3, whee g53.0 ad/s and b ad/s 3. (a) Calculate the angula velocit of the olle as a function of time. (b) Calculate the angula acceleation of the olle as a function of time. (c) What is the maimum positive angula velocit, and at what value of t does it occu? *9.67. A disk of adius 5.0 cm is fee to tun about an ale pependicula to it though its cente. It has ve thin but stong sting wapped aound its im, and the sting is attached to a ball that is pulled tangentiall awa fom the im of the disk (Fig. 9.33). The pull inceases in magnitude and poduces an acceleation of the ball that obes the equation a t 5 At, whee t is in seconds and A is a constant. The clinde stats fom est, and at the end of the thid second, the ball s acceleation is.80 m/s. (a) Find A. (b) Epess the angula acceleation of the disk as a function of time. (c) How much time afte the disk has begun to tun does it each an angula speed of 5.0 ad/s? (d) Though what angle has the disk tuned just as it eaches 5.0 ad/s? (Hint: ee ection.6.) Figue 9.33 oblem Disk Ball ull When a to ca is apidl scooted acoss the floo, it stoes eneg in a flwheel. The ca has mass 0.80 kg, and its flwheel has moment of inetia kg # m. The ca is 5.0 cm long. An advetisement claims that the ca can tavel at a scale speed of up to 700 km/h 440 mi/h. The scale speed is the speed of the to ca multiplied b the atio of the length of an actual ca to the length of the to. Assume a length of 3.0 m fo a eal ca. (a) Fo a scale speed of 700 km/h, what is the actual tanslational speed of the ca? (b) If all the kinetic eneg that is initiall in the flwheel is conveted to the tanslational kinetic eneg of the to, how much eneg is oiginall stoed in the flwheel? (c) What initial angula velocit of the flwheel was needed to stoe the amount of eneg calculated in pat (b)? A classic 957 Chevolet Covette of mass 40 kg stats fom est and speeds up with a constant tangential acceleation of 3.00 m/s on a cicula test tack of adius 60.0 m. Teat the ca as a paticle. (a) What is its angula acceleation? (b) What is its angula speed 6.00 s afte it stats? (c) What is its adial acceleation at this time? (d) ketch a view fom above showing the cicula tack, the ca, the velocit vecto, and the acceleation component vectos 6.00 s afte the ca stats. (e) What ae the magnitudes of the total acceleation and net foce fo the ca at this time? (f) What angle do the total acceleation and net foce make with the ca s velocit at this time? Enginees ae designing a sstem b which a falling mass m impats kinetic eneg to a otating unifom dum to which it is attached b thin, ve light wie wapped aound the im of the dum (Fig. 9.34). Thee is no appeciable fiction in the ale of the dum, and evething stats fom est. This sstem is being tested on eath, but it is to be used on Mas, whee the acceleation Figue 9.34 oblem Dum due to gavit is 3.7 m/s. In the m eath tests, when m is set to 5.0 kg and allowed to fall though 5.00 m, it gives 50.0 J of kinetic eneg to the dum. (a) If the sstem is opeated on Mas, though what distance would the mass have to fall to give the same amount of kinetic eneg to the dum? (b) How fast would the 5.0-kg mass be moving on Mas just as the dum gained 50.0 J of kinetic eneg? 9.7. A vacuum cleane belt is looped ove a shaft of adius 0.45 cm and a wheel of adius.00 cm. The aangement of the belt, shaft, and wheel is simila to that of the chain and spockets in Fig The moto tuns the shaft at 60.0 ev/s and the moving belt tuns the wheel, which in tun is connected b anothe shaft to the olle that beats the dit out of the ug being vacuumed. Assume that the belt doesn t slip on eithe the shaft o the wheel. (a) What is the speed of a point on the belt? (b) What is the angula velocit of the wheel, in ad/s? 9.7. The moto of a table saw is otating at 3450 ev/min. A pulle attached to the moto shaft dives a second pulle of half the diamete b means of a V-belt. A cicula saw blade of diamete 0.08 m is mounted on the same otating shaft as the second pulle. (a) The opeato is caeless and the blade catches and thows back a small piece of wood. This piece of wood moves with linea speed equal to the tangential speed of the im of the blade. What is this speed? (b) Calculate the adial acceleation of points on the oute edge of the blade to see wh sawdust doesn t stick to its teeth A wheel changes its angula velocit with a constant angula acceleation while otating about a fied ais though its cente. (a) how that the change in the magnitude of the adial acceleation duing an time inteval of a point on the wheel is twice the poduct of the angula acceleation, the angula displacement, and the pependicula distance of the point fom the ais. (b) The adial acceleation of a point on the wheel that is 0.50 m fom the ais changes fom 5.0 m/s to 85.0 m/s as the wheel otates though 5.0 ad. Calculate the tangential acceleation of this point. (c) how that the change in the wheel s kinetic eneg duing an time inteval is the poduct of the moment of inetia about the ais, the angula acceleation, and the angula displacement. (d) Duing the 5.0-ad angula displacement of pat (b), the kinetic eneg of the wheel inceases fom 0.0 J to 45.0 J. What is the moment of inetia of the wheel about the otation ais? A sphee consists of a solid wooden ball of unifom densit 800 kg/m 3 and adius 0.0 m and is coveed with a thin coating of lead foil with aea densit 0 kg/m. Calculate the moment of inetia of this sphee about an ais passing though its cente Estimate ou own moment of inetia about a vetical ais though the cente of the top of ou head when ou ae standing up staight with ou ams outstetched. Make easonable appoimations and measue o estimate necessa quantities A thin unifom od 50.0 cm long with mass 0.30 kg is bent at its cente into a V shape, with a 70.0 angle at its vete. Find the moment of inetia of this V-shaped object about an ais pependicula to the plane of the V at its vete It has been agued that powe plants should make use of offpeak hous (such as late at night) to geneate mechanical eneg and stoe it until it is needed duing peak load times, such as the middle of the da. ne suggestion has been to stoe the eneg in lage flwheels spinning on neal fictionless ball beaings. Conside a flwheel made of ion (densit 7800 kg/m 3 ) in the shape of a 0.0-cm-thick unifom disk. (a) What would the diamete of such a disk need to be if it is to stoe 0.0 megajoules of kinetic eneg when spinning at 90.0 pm about an ais pependicula to the disk at its cente? (b) What would be the centipetal acceleation of a point on its im when spinning at this ate? While edesigning a ocket engine, ou want to educe its weight b eplacing a solid spheical pat with a hollow spheical shell of the same size. The pats otate about an ais though thei cente. You need to make sue that the new pat alwas has the same otational kinetic eneg as the oiginal pat had at an given ate of otation. If the oiginal pat had mass M, what must be the mass of the new pat? The eath, which is not a unifom sphee, has a moment of inetia of MR about an ais though its noth and south poles. It takes the eath 86,64 s to spin once about this ais. Use Appendi F to calculate (a) the eath s kinetic eneg due to its otation about this ais and (b) the eath s kinetic eneg due to its obital motion aound the sun. (c) Eplain how the value of the eath s moment of inetia tells us that the mass of the eath is concentated towad the planet s cente A unifom, solid disk with mass m and adius R is pivoted about a hoizontal ais though its cente. A small object of the same mass m is glued to the im of the disk. If the disk is eleased fom est with the small object at the end of a hoizontal adius, find the angula speed when the small object is diectl below the ais A metal sign fo a ca dealeship is a thin, unifom ight tiangle with base length b and height h. The sign has mass M. (a) What is the moment of inetia of the sign fo otation about the side of length h? (b) If M kg, b 5.60 m, and h 5.0 m, what is the kinetic eneg of the sign when it is otating about an ais along the.0-m side at.00 ev/s? 9.8. Measuing I. As an inten with an engineeing fim, ou ae asked to measue the moment of inetia of a lage wheel, fo otation about an ais though its cente. ince ou wee a good phsics student, ou know what to do. You measue the diamete of the wheel to be m and find that it weighs 80 N. You mount the wheel, using fictionless beaings, on a hoizontal ais though the wheel s cente. You wap a light ope aound the wheel and hang a 8.00-kg mass fom the fee end of the ope, as shown in Fig You elease the mass fom est; the mass descends and the wheel tuns as the ope unwinds. You find that the mass has speed 5.00 m/s afte it has descended.00 m. (a) What is the moment of inetia of the wheel fo an ais pependicula to the wheel at its cente? (b) You boss tells ou that a lage I is needed. He asks ou to design a wheel of the same mass and adius that has I kg # m. How do ou epl? A mete stick with a mass of 0.60 kg is pivoted about one end so it can otate without fiction about a hoizontal ais. The mete stick is held in a hoizontal position and eleased. As it swings though the vetical, calculate (a) the change in gavitational potential eneg that has occued; (b) the angula speed of the stick; (c) the linea speed of the end of the stick opposite the ais. (d) Compae the answe in pat (c) to the speed of a paticle that has fallen.00 m, stating fom est Eactl one tun of a fleible ope with mass m is wapped aound a unifom clinde with mass M and adius R. The clinde otates without fiction about a hoizontal ale along the clinde ais. ne end of the ope is attached to the clinde. The clinde stats with angula speed v 0. Afte one evolution of the clinde the ope has unwapped and, at this instant, hangs veticall down, tangent to the clinde. Find the angula speed of the clinde and the linea speed of the lowe end of the ope at this time. You can ignoe the thickness of the ope. [Hint: Use Eq. (9.8).] The pulle in Fig has adius R and a moment of inetia I. The ope does not slip ove the pulle, and the pulle spins on a fictionless ale. The coefficient of kinetic fiction between block A and the tabletop is m k. The sstem is eleased fom est, and block B descends. Block A has mass m A and block B has mass m B. Use eneg methods to calculate the speed of block B as a function of the distance d that it has descended. Figue 9.35 oblem A B I

16 34 CHATER 9 Rotation of Rigid Bodies Challenge oblems The pulle in Fig has Figue 9.36 oblem adius 0.60 m and moment of inetia kg # m. The ope does not slip on the pulle im. Use eneg methods to calculate the speed of the 4.00-kg block 4.00 kg just befoe it stikes the floo You hang a thin hoop with adius R ove a nail at the im of 5.00 m the hoop. You displace it to the side (within the plane of the hoop) though an angle b fom its equilibium position and let it go..00 kg What is its angula speed when it etuns to its equilibium position? [Hint: Use Eq. (9.8).] A passenge bus in Zuich, witzeland, deived its motive powe fom the eneg stoed in a lage flwheel. The wheel was bought up to speed peiodicall, when the bus stopped at a station, b an electic moto, which could then be attached to the electic powe lines. The flwheel was a solid clinde with mass 000 kg and diamete.80 m; its top angula speed was 3000 ev/min. (a) At this angula speed, what is the kinetic eneg of the flwheel? (b) If the aveage powe equied to opeate the bus is W, how long could it opeate between stops? Two metal disks, one with adius Figue 9.37 R 5.50 cm and mass M kg and oblem the othe with adius R cm and mass M 5.60 kg, ae welded togethe and mounted on a fictionless ais though R thei common cente (Fig. 9.37). (a) What R is the total moment of inetia of the two disks? (b) A light sting is wapped aound the edge of the smalle disk, and a.50-kg block is suspended fom the fee end of the sting. If the block is eleased fom est at a distance of.00 m above the floo, what is its speed just befoe it stikes the floo? (c) Repeat the calculation of pat (b), this time with the sting wapped aound the edge of the lage disk. In which case is the.50 kg final speed of the block the geatest? Eplain wh this is so In the clinde and mass combination descibed in Eample 9.9 (ection 9.4), suppose the falling mass m is made of ideal ubbe, so that no mechanical eneg is lost when the mass hits the gound. (a) If the clinde is oiginall not otating and the mass m is eleased fom est at a height h above the gound, to what height will this mass ebound if it bounces staight back up fom the floo? (b) Eplain, in tems of eneg, wh the answe to pat (a) is less than h In the sstem shown in Fig. 9.8, a.0-kg mass is eleased fom est and falls, causing the unifom 0.0-kg clinde of diamete 30.0 cm to tun about a fictionless ale though its cente. How fa will the mass have to descend to give the clinde 50 J of kinetic eneg? 9.9. In Fig. 9.38, the clinde and pulle tun without fiction about stationa hoizontal ales that pass though thei centes. A light ope is wapped aound the clinde, passes ove the pulle, and has a 3.00-kg bo suspended fom its fee end. Thee is no slip- Figue 9.38 oblem 9.9. Clinde ulle Bo ping between the ope and the pulle suface. The unifom clinde has mass 5.00 kg and adius 40.0 cm. The pulle is a unifom disk with mass.00 kg and adius 0.0 cm. The bo is eleased fom est and descends as the ope unwaps fom the clinde. Find the speed of the bo when it has fallen.50 m A thin, flat, unifom disk has mass M and adius R. A cicula hole of adius R/4, centeed at a point R/ fom the disk s cente, is then punched in the disk. (a) Find the moment of inetia of the disk with the hole about an ais though the oiginal cente of the disk, pependicula to the plane of the disk. (Hint: Find the moment of inetia of the piece punched fom the disk.) (b) Find the moment of inetia of the disk with the hole about an ais though the cente of the hole, pependicula to the plane of the disk A pendulum is made of a unifom solid sphee with mass M and adius R suspended fom the end of a light od. The distance fom the pivot at the uppe end of the od to the cente of the sphee is L. The pendulum s moment of inetia I fo otation about the pivot is usuall appoimated as ML. (a) Use the paallel-ais theoem to show that if R is 5% of L and the mass of the od is ignoed, I is onl 0.% geate than ML. (b) If the mass of the od is % of M and R is much less than L, what is the atio of I od fo an ais at the pivot to ML? ependicula-ais Theoem. Conside a igid bod that is a thin, plane sheet of abita shape. Take the bod to lie in the -plane and let the oigin of coodinates be located at an point within o outside the bod. Let I and I be the moments of inetia about the - and -aes, and let I be the moment of inetia about an ais though pependicula to the plane. (a) B consideing mass elements m i with coodinates i, i, show that I I 5 I. This is called the pependicula-ais theoem. Note that point does not have to be the cente of mass. (b) Fo a thin washe with mass M and with inne and oute adii R and R, use the pependicula-ais theoem to find the moment of inetia about an ais that is in the plane of the washe and that passes though its cente. You ma use the infomation in Table 9.. (c) Use the pependiculaais theoem to show that fo a thin, squae sheet with mass M and side L, the moment of inetia about an ais in the plane of the sheet that passes though the cente of the sheet is ML. You ma use the infomation in Table A thin, unifom od is bent into a squae of side length a. If the total mass is M, find the moment of inetia about an ais though the cente and pependicula to the plane of the squae. (Hint: Use the paallel-ais theoem.) *9.97. A clinde with adius R and mass M has densit that inceases lineal with distance fom the clinde ais, 5a, whee a is a positive constant. (a) Calculate the moment of inetia of the clinde about a longitudinal ais though its cente in tems of M and R. (b) Is ou answe Figue 9.39 oblem geate o smalle than the moment of inetia of a clinde of the same mass and adius but of unifom densit? Eplain wh this esult makes qualitative sense Neuton tas and upenova Remnants. The Cab Nebula is a cloud of glowing gas about 0 light-eas acoss, located about 6500 light eas fom the eath (Fig. 9.39). It is the emnant of a sta that undewent a supenova eplosion, seen on eath in 054 A.D. Eneg is eleased b the Cab Nebula at a ate of about W, about 0 5 times the *9.00. Calculate the moment of inetia of a unifom solid cone about an oblem Figue 9.40 Challenge ate at which the sun adiates eneg. The Cab Nebula obtains its eneg fom the otational kinetic eneg of a apidl spinning ais though its cente (Fig. 9.40). neuton sta at its cente. This object otates once eve s, The cone has mass M and altitude h. and this peiod is inceasing b R s fo each second of The adius of its cicula base is R. time that elapses. (a) If the ate at which eneg is lost b the neuton sta is equal to the ate at which eneg is eleased b the neb- is coded in a patten of tin pits 9.0. n a compact disc (CD), music ula, find the moment of inetia of the neuton sta. (b) Theoies of aanged in a tack that spials outwad towad the im of the disc. As h supenovae pedict that the neuton sta in the Cab Nebula has a mass about.4 times that of the sun. Modeling the neuton sta as a the disc spins inside a CD plae, the solid unifom sphee, calculate its adius in kilometes. (c) What is tack is scanned at a constant linea the linea speed of a point on the equato of the neuton sta? Compae to the speed of light. (d) Assume that the neuton sta is uni- adius of the tack vaies as it spials Ais speed of v 5.5 m/s. Because the fom and calculate its densit. Compae to the densit of odina outwad, the angula speed of the ock 3000 kg/m 3 and to the densit of an atomic nucleus (about disc must change as the CD is plaed. (ee Eecise 9..) Let s 0 7 kg/m 3 ). Justif the statement that a neuton sta is essentiall see what angula acceleation is equied to keep v constant. The a lage atomic nucleus. equation of a spial is u 5 0 bu, whee 0 is the adius of the spial at u50 and b is a constant. n a CD, 0 is the inne adius of the spial tack. If we take the otation diection of the Challenge oblems CD to be positive, b must be positive so that inceases as the disc The moment of inetia of a sphee with unifom densit tuns and u inceases. (a) When the disc otates though a small about an ais though its cente is 5 MR MR. atellite angle du, the distance scanned along the tack is ds 5 du. Using obsevations show that the eath s moment of inetia is MR. the above epession fo u, integate ds to find the total distance s scanned along the tack as a function of the total angle u Geophsical data suggest the eath consists of five main egions: the inne coe 5 0 to 5 0 km of aveage densit though which the disc has otated. (b) ince the tack is scanned,900 kg/m 3, the oute coe 5 0 km to km of at a constant linea speed v, the distance s found in pat (a) is aveage densit 0,900 kg/m 3, the lowe mantle km to equal to vt. Use this to find u as a function of time. Thee will be km of aveage densit 4900 kg/m 3, the uppe mantle two solutions fo u ; choose the positive one, and eplain wh this km to km of aveage densit 3600 kg/m 3, is the solution to choose. (c) Use ou epession fo u t to find and the oute cust and oceans km to km of the angula velocit v z and the angula acceleation a z as functions of time. Is a z constant? (d) n a CD, the inne adius of the aveage densit 400 kg/m 3. (a) how that the moment of inetia about a diamete of a unifom spheical shell of inne adius R, tack is 5.0 mm, the tack adius inceases b.55 mm pe evolution, and the plaing time is 74.0 min. Find the values of 0 and oute adius R and densit is I 58p/5 R 5 R 5,. (Hint: Fom the shell b supeposition of a sphee of densit and a b, and find the total numbe of evolutions made duing the plaing time. (e) Using ou esults fom pats (c) and (d), make smalle sphee of densit.) (b) Check the given data b using them to calculate the mass of the eath. (c) Use the given data to gaphs of v (in ad/s) vesus t and a (in ad/s z z ) vesus t between calculate the eath s moment of inetia in tems of MR. t 5 0 and t min.