Chapte 7 Rotational Motion and the Law of Gaity What is a Rigid Body? Rotational Kinematics Angula Velocity ω and Acceleation α Unifom Rotational Motion: Kinematics Unifom Cicula Motion: Kinematics and Dynamics Applications Newton s Law of Uniesal Gaitational Attaction Planetay motion
Rigid Body and its Motions Until now we consideed only the motion of point-like objects. Objects with extended size can be consideed as a collection of many point-like paticles. When these paticles do not moe with espect to each othe, the system is called a igid body: it cannot be defomed. The geneal motion of a igid body can be split into two types: Tanslational (Linea) Motion Rotational (Angula) Motion we need an angula fomalism Object and teminology:,,, a, F, p...,,,,, L... Linea elocity Rigid body A paticle of the body Tanslation elocity Cente o axis of otation
Angula Kinematics Angula Position θ In puely otational motion, all points on the object moe in cicles aound the axis of otation ( O ), with each point descibed by a ecto position Def: The angle θ made by the position ecto with espect to an abitay axis (say x) is called angula position θ O y θ x Ex: see the two points on the adjacent bicycle wheel: notice that, as long they ae not on the same adius, the points on the igid body will hae diffeent angula positions In ou appoach to otations, angles will be measued in adians: 1 adian is the angle at the cente of a cicle subtending an ac equal to the adius of the cicle When the angle at the cente is expessed in adians, the length of the ac subtended is gien by: l Ex: The cicumfeence π of a complete otation subtends an angle of π Conention: angles measued counteclockwise ae positie angle measued clockwise ae negatie + l = θ θ = 1 ad l = θ θ
Angula Kinematics Angula Displacement and Velocity How can we use angula positions to descibe otations? Notice that een though the angula positions of diffeent points of a wheel ae in geneal diffeent, when the wheel otates, all points otate though the same angle Def: The change in angula position of all the points on a otating igid body is called angula displacement: 1 Δθ θ θ1 Abitay adius x Then, if we want to efe to how fast the angula position changes we hae to define fist the aeage angula elocity as the angula displacement diided by time: 1 Theefoe, like in the linea case, the instantaneous angula elocity is gien by: lim t 0 t 1 t t t Ex: If the bicycle wheel makes two complete otations eey second, we say that it has a constant angula speed of (π/1 s) = 4π ad/s SI ad s
Angula Kinematics Angula Acceleation. Angula ecto diections Hence we can define the aeage angula acceleation as the ate at which the angula elocity changes with time: 1 t t t 1 lim t 0 t SI ad s Hence, the instantaneous angula acceleation: The diection of angula elocity is gien by a ight hand ule 0 0 Ex: If the bicycle wheel spins each second though an angle lage and lage by π, we say that each complete otation its aeage angula elocity is π, so its instantaneous angula elocity inceases by π ad/s, so it has a constant angula acceleation π ad/s Although it is not as intuitie as in the tanslational case, the angula elocity and acceleation ae ectos, pependicula on the cicle of otation: The diection of angula acceleation is paallel o antipaallel with the angula elocity depending on whethe ω inceases o deceases
Relating Linea and Angula Kinematics So, the angula displacement, elocity and acceleation chaacteize the entie igid body: all points hae the same Δθ, ω and α Howee, the linea distance, elocity and acceleation of points at aious adii fom the axis of otation ae diffeent: each point has a diffeent Δl, and a The linea kinematics of each point on a igid body can be elated to the oeall angula chaacteistics based on the elationship l = θ Fo instance, conside a wheel otating with constant angula speed ω. A point at distance fom the cente of otation will otate with constant linea speed taeling an ac Δl in a time Δt: theefoe, we obtain l t t So, as long as they ae not at the same distance fom the cente of otation, the diffeent points on a igid body hae diffeent linea speeds, inceasing fom zeo in the cente of otation to a maximum alue on the oute im of the otating igid body 3 = ω 3 = ω Δl = Δθ 1 = ω 1 Δθ 3 1 ω ω x
How about Acceleation? The acceleation is a bit moe complex since in geneal the ecto linea acceleation of a paticle in cicula motion is not tangent to the tajectoy. Howee, it can be consideed as haing two components one tangent to the tajectoy (paallel o anti-paallel with the elocity) and one pependicula on the elocity: a t : tangent, descibes how the magnitude of the linea elocity aies a : adial (o centipetal), descibes how the diection of the elocity aies The component at of a paticle at distance fom the axis of otation can be easily elated to the angula acceleation α of the igid body: a a t a t t t a x The component a (also called centipetal since it always point towad the cente of otation) makes necessay a sepaate discussion a few slides futhe ω
Execise : Components of acceleation A paticle moes as shown in the figue. Between points B and D, the path is a staight line. Let s figue out the net acceleation ectos in points A, C and E along the path epesented below, fo each of the following cases: a) the paticle moes with steadily constant speed b) the paticle moes with steadily inceasing speed c) the paticle moes with steadily deceasing speed
Angula Kinematics Unifomly acceleated otation The linea-angula elationship poides an easy way to descibe cicula motion with constant angula acceleation α, by simply noticing that each point on the otating igid body acceleates unifomly along the espectie ac of cicle Assume that the motion stats at t 0 = 0 when the otation is chaacteized by θ 0, ω 0 0 ω 0 t 0 = 0 At time t ω Δl x Δθ x Then, if the angula acceleation α is constant, at a late instant t, The linea motion of one paticle of the igid body at distance fom the cente of otation l t a t 1 0 a t 0 a l 0 1 0 l t t t t if each linea quantity is diided by, we obtain t t t t 0 1 0 0 t 0 1 0 0 t The otational motion of the entie igid body (alid fo any of its pats)
Obseation useful in poblems: Rolling Motion (Without Slipping) In figue (a), if the wheel of adius is olling without slipping, the point P on the im is at est with espect to the floo when it touches it, while the cente C moes with elocity to the ight In figue (b), the same wheel is seen fom a efeence fame whee C is at est. Now point P is moing with elocity. Since P is a point at distance fom the cente of otation, we hae: Theefoe, when a wheel, o a sphee, o a cylinde olls, its tanslational speed (that is, the speed cm of its cente of mass) is elated to its angula speed ω by cm a) Wheel seen by someone on the gound: ω C P b) Wheel seen by someone on the bike: ω C Caution: een though this has the same fom as the elation between linea speed of a point and the angula speed, it is not the same thing. P
Poblems: 1. Unifomly acceleated otation: An automobile engine slows down fom 4500 pm to 500 pm in.5 s. Calculate a) its angula acceleation (assumed constant) b) the total numbe of eolutions the engine makes in this time. 4500 0 4500 pm ad 60 s 500 1 500 pm ad 60 s s 150 ad s 83 ad. Rotational and linea motion: You ae to design a otating cylindical axle to lift buckets of cement fom the gound to a ooftop. The buckets will be attached to a hook on the fee end of a cable that waps aound the im of the axle; as the axle tuns, the buckets will ise. a) What should the diamete of the axle be in ode to aise the buckets at a steady.00 cm/s when it is tuning at 7.5 pm? b) If instead the axle must gie the buckets an upwad acceleation of 0.400 m/s, what should be the angula acceleation of the axle be? 7.5 7.5 pm ad s ad s 60 s 4 s s
Centipetal Acceleation So, the adial (o centipetal) acceleation descibes how the diection of the elocity changes: that is, if the object moes in a staight line, a = 0 To see how a is elated to the speed, we ll conside a paticle moing in a cicle of adius with α = 0, that is, a constant speed (caution! the elocity is not constant) Assume that the paticle taels an ac-distance Δl subtending an angula displacement Δθ in a time Δt Then, looking at the coesponding change in elocity of the eoling paticle and using the definition of acceleation, we find that the centipetal acceleation is gien by This expession is alid fo any paticle moing along a cued tajectoy: if the cuatue of the path can be fitted locally by a cicle of adius and the instantaneous speed is, the expession aboe gies the adial component of the acceleation in the espectie point a 1 1 ~Δl Δθ Two simila tiangles fom, such that: l l a l t t a a 1 1 1 1 a a path path
T Some useful quantities and a summay We can intoduce a new set of paametes descibing peiodic motion: 1. Fequency f : the numbe of eolutions pe time. Peiod T: time equied to complete a eolution Fequency and peiod ae the inese of each othe: Summaizing list of coespondences between linea and otational quantities: Linea Type Rotational Relation x displacement θ x = θ elocity ω = ω a t acceleation α a t = α a acceleation - a = ω f 1 T Using the idea of peiod we can see once moe how the elationship between the linea elocity and the angula elocity makes sense and anothe elationship fo a : a f 1 s Hetz, Hz SI T SI s
Unifom Cicula Motion Kinematics The unifom cicula motion is the motion of a paticle in a cicle of constant adius at constant speed, such that α is zeo Being always tangent to the cicula path the ecto instantaneous elocity changes diection, albeit its magnitude stays constant Hence, the tangential acceleation a t is zeo, and the net acceleation is gien only by the centipetal acceleation a pointing eeywhee pependicula on the elocity Comments: Physical situation: a paticle moing in a cicle: The magnitude of the centipetal acceleation is lage if the speed is lage The magnitude of the centipetal acceleation is lage if the adius of otation is small path a Constant speed Ex: if a ca takes a tun at high speed, it will hae a lage centipetal acceleation than when taking it slowly Ex: if a ca takes a tun shap tun (small adius), it will hae a lage centipetal acceleation than when taking a wide tun a
By Newton s nd Law, since in the cicula motion the acceleation is necessaily not zeo (since the elocity must change in diection), we see that fo an object to be in unifom cicula motion thee must be a net foce acting on it. We aleady know the acceleation, so can immediately wite the foce: F ma m This centipetal foce is not a new foce: any net foce pointing adially inwad the cicula tajectoy (pependicula on the elocity) can play the ole of centipetal foce, since it has as a esult a change in diection of elocity Comments: Unifom Cicula Motion Dynamics A common misconception is to assume that an object on a cued tajectoy is thown out of it by an outwad centifugal foce. We now see that the foce must be actually inwad The objects taking tuns ae appaently pushed outwad by thei inetia, while the centipetal foce keeps it on the tajectoy If the centipetal foce anishes, the object flies off tangent to the cicle, not outwad as if a centifugal foce wee pesent path F
Execise 1: How Angelina Succumbed to Bad Physics In the moie Wanted, bullets ae cued by skilled tattooed assassins. Fo instance, Angelina kills heself by fiing a bullet in a cicle passing though the skulls of some bald dudes befoe hitting he. Say that the bullet has a mass m = 4. g and a muzzle speed of 600 m/s. Also, say that the dudes aanged themseles coneniently in a cicle with adius = 5.0 m. a) How big should be the centipetal foce keeping the bullet on the cicula tajectoy? Meditate about the possible oigin of such a foce and how ealistic is such a scenaio b) How fast should Angelina toss the gun to the sweaty guy in the middle fo the scene to make sense?
Execise : Ball eoling as held by a sting A ball of mass m is connected by a sting of length and moed into a cicula tajectoy with constant speed. Let s estimate the foce a peson must exet on the sting to make the ball eole in a hoizontal cicle. T a) What is the natue of the centipetal foce exeted on the ball? b) Based on Newton s nd Law, what is this foce in tems of gien quantities? Poblem 3. Tension as a centipetal foce: Now let s assume that the ball fom the execise aboe is swung in a etical cicle, still with a constant speed. a) Detemine the tension in an abitay point of the cicle whee the adius makes an angle θ with espect to the hoizontal b) Use the esult fom pat (a) to detemine the tension in the sting when the sting is hoizontal, on top of the cicle, and at the bottom of the cicle.
Poblems: 4. Nomal as a centipetal foce: A small emote-contol ca with mass m = 1.60 kg moes at a constant speed of = 1.0 m/s in a etical cicle inside a hollow metal cylinde that has a adius of = 5.00 m. What is the magnitude of the nomal foce exeted on the ca by the walls of the cylinde at a) an abitay point on the loop b) point A (bottom of the etical cicle) c) point B (top of the etical cicle) 5. Conical pendulum: A bob of mass m is suspended fom a fixed point with a massless sting of length L (i.e., it is a pendulum). What tangential speed must the bob hae so that it moes in a hoizontal cicle with the sting always making an angle θ fom the etical?
Nonunifom Cicula Motion Elementay obseations If the elocity of a eoling paticle changes in magnitude as well, the objects is said to be in a nonunifom cicula motion As any eoling object, the object has a centipetal acceleation: a Howee, in this case, the paticle also has a tangential acceleation: at Physical situation: a paticle moing in a cicle: path a a Changing speed a t Hence, at a cetain moment when the instantaneous angula speed is ω, the instantaneous net acceleation of the paticle has a magnitude gien by: a a a 4 t Accodingly, the net foce acting on the object will contibute to the change in elocity diection with a centipetal component, and to the change in speed with a tangential component
Newton s Law of Uniesal Gaitation The idea We leaned that Eath exets a foce on any mass the weight. We studied how the weight contibutes to the motion of the object, and we quantified its stength though its effect when it acts alone: the gaitational acceleation g Howee, what is this foce? Fo instance, is it acted only by Eath? Quiz: Based on what law can we infe that the gaitational foce is not acted only by Eath? The fomulation of a coheent theoy of gaity (consistent with his mechanics) is anothe example of Newton s many fundamental contibutions to Physics He ealized that gaity acts between any masses: it is just logical to assume that the downwad foce that pulls onto an apple o a peson must be hae the same natue as the attaction exeted by Eath on the Moon in ode to keep it on its obit Moeoe, by studying the motion of the planets, he deduced that the magnitude of the gaitational attaction must ay inesely popotional to the squae of the distance between the inteacting masses weight
Eey two paticles in the uniese attact each othe with gaitational foces diectly popotional to the poduct of thei masses m and M, and inesely popotional to the squae of the distance between them. Comments: Newton s Uniesal Law of Gaitation Quantitatiely Fg G mm Gaitational constant G 6.67310 Nm kg 11 The foce of attaction is along the line connecting the centes of gaity of the objects (which in the cases we ae inteested in coincides with thei centes of mass o symmety) Ex: Two paticles: m F g F g M Hence, the distance between objects is the distance between thei centes of gaity Note the pesence of Newton s 3 d law in this law: mass m attacts mass M and ice-esa with action-eaction foces Two sphees: m Two squae plates: M The gaitational constant G is ey small, meaning that the gaitational attaction will be hadly obseable between small masses m M
Newton s Uniesal Law of Gaitation Detemining G The fist detemination of the alue of the gaitational constant G is sometimes attibuted (wongly) to Si Heny Caendish Caendish actually tied to find expeimentally the density of Eath. His data was only much late used to calculate G Caendish s appaatus was based on the obseation of the gaitational attaction between elatiely small objects using a tosion balance If the foce of attaction is known, G can be calculated easily Moden esion of Caendish s expeimental aangement:
Poblems: 6. Pinciple of supeposition: Thee identical paticles of mass m ae positioned at thee cones of a isosceles ight tiangle of equal sides a, as in the figue. Calculate the total gaitational attaction on mass 1. m 1 a a m a 3 m
Gaitational Acceleation In a moe geneic context Until now, we assumed the local gaitational acceleation constant. Howee, wee see that this is just an appoximation sing the weight does depend on the distance to the cente of attaction (Eath): h m mg h mg m g h G mme me me 1 g h G G h 1h E E E E On the suface of the Eath h 0 so m E mme me mg ~ 9.8 m s h0 G g G E E Theefoe, the gaitational acceleation at altitude h aboe the suface of the Eath can be witten: g h g 9.8 m s 1h 1h E So, the acceleation due to gaity aies oe the Eath s suface due to altitude, local geology, and the shape of the Eath, which is not quite spheical. E
Applications of the Law of Gaity Satellites The motion of a satellite can be associated with a pojectile tajectoy missing the suface of the Eath due to the high speed. If the speed is too small, the satellite falls on the suface (paths 1-) In ode to put a satellite on the obit (paths 3-5), it must hae a minimum speed called satellite speed, which can be estimated assuming a cicula obital tajectoy of adius. Then mm E 1 mg ma G m Howee, if the initial speed is too geat, the satellite flies out into space and becomes a spacecaft as it flies along open (paths 6-7) The minimum speed necessay fo the object to escape the Eath s gaity is called escape speed and can be estimated by noticing that ey fa away fom eath the gaitational potential enegy anishes. Theefoe mm m E E 0 0 G 1 KEinitial PE PE E m G E 1 m G E m E Launching towe E E
Applications of the Law of Gaity Keple s Laws 1. The path of each planet about the Sun is an ellipse, with the Sun at one focus. Comment: This behaio is due to the specific 1/ dependence of the foce on the distance.. The position of each planet about the Sun sweeps out equal aeas in equal times. Comment: This behaio is explained by the conseation of angula momentum (we ll lean about it in the next chapte) 3. The peiods of the planets ae popotional to 3/ powes of the semi-majo axis lengths of thei obits. Comment: This is explained by Newton s nd Law applied to the obiting body. Fo any two planets, T T a a 1 1 3