TOPC D: OTTON SPNG 08. ngula kinematics. ngula velocit and angula acceleation. Constant-angula-acceleation fomulae. Displacement, velocit and acceleation in cicula motion. ngula dnamics. Toque. ngula momentum. The angula-momentum pinciple fo motion in a cicle.4 The angula-momentum pinciple fo abita motion. igid-bod otation. Moment of inetia. Second moments and the adius of gation. The equation of otational motion.4 Compaing tanslation and otation.5 Eamples 4. Calculation of moments of inetia 4. Methods of calculation 4. Fundamental shapes 4. Stetching paallel to an ais 4.4 Volumes of evolution 4.5 Change of ais 5. Geneal motion of a igid bod (optional) 5. olling without slipping 5. olling with slipping ppendi : Moments of inetia of simple shapes ppendi : Second moment of aea Mechanics Topic D (otation) - David psle
. NGUL KNEMTCS. ngula Velocit and ngula cceleation P Fo a paticle moving in a cicula ac, o fo a igid bod otating about a fied ais, the instantaneous position is defined b the angle between a adius vecto and a fied line. O s f s is length of ac and is adius then the angle θ in adians is defined such that s θ () ngula velocit ω is the ate of change of angle: dθ ω () ngula acceleation α is the ate of change of angula velocit: dω d θ α () t is common to use a dot to indicate diffeentiation w..t. time; e.g. θ means dθ/.. Constant-ngula-cceleation Fomulae Thee is a diect coespondence between linea and angula motion. Linea ngula Displacement s θ Velocit cceleation Constant-acceleation fomulae ds v dv d s dv a v ds s ( u v) t v u at s ut at v u as α dθ ω dω d θ (ω θ 0 ω)t ω ω 0 αt θ ω 0 t αt ω ω 0 αθ dω ω dθ Fo non-constant acceleation: distance is the aea unde a v t gaph; angle is the aea unde an ω t gaph; v acceleation is the gadient of a v t gaph; angula acceleation is the gadient of a ω t gaph. s t t Mechanics Topic D (otation) - David psle
Eample. What is the angula velocit in adians pe second of the minute hand of a clock? Eample. tubine stats fom est and has a constant angula acceleation of 0. ad s. How man evolutions does it make in eaching a otation ate of 50 pm?. Displacement, Velocit and cceleation in Cicula Motion Conside a paticle moving at a fied adius. The following have alead been deived in Topic (Kinematics), as a special case of motion in geneal pola coodinates. v Velocit Since s = θ and is constant, the velocit is tangential and its magnitude (speed) is dθ v ω (4) cceleation Because it is not moving in a staight line, the paticle has two components of acceleation: dv v tangential, if its speed is changing: dv dω o d t O (5a) adiall inwad, because its diection is changing: v o ω (5b) The latte is called the centipetal acceleation. centipetal foce is necessa to maintain this and keep the paticle moving in a cicula path. This foce can be povided in man was: fo eample, the tension in a cable, a nomal eaction fom an oute bounda o fiction. Mechanics Topic D (otation) - David psle
Eample. ca entes a cicula cuve of adius 50 m at a stead speed of 50 km h. f thee is no banking, what coefficient of fiction is necessa between the tes and the oad to pevent the ca skidding? Eample 4. Find the minimum coefficient of fiction necessa to pevent slipping fo a paticle which is placed: (a) (b) 00 mm fom the otation ais on a tuntable otating at 78 pm; on the inside of a clindical dum, adius 0. m, otating about a vetical ais at 00 pm. Eample 5. (Eam 07) paticle of mass kg is whiled aound in a hoizontal cicle b a light elastic sting of unstetched length.5 m and stiffness 90 N m attached to a fied point O. t a paticula speed, the cable makes an angle of 5º with the hoizontal. Find: (a) the tension in the cable; (b) the etension of the cable; (c) the speed of the paticle. O 5 o kg Mechanics Topic D (otation) - 4 David psle
Eample 6. (Eam 00) building s oof consists of a smooth hemispheical dome with outside adius 0 m. bief gust of wind dislodges a small object at the top of the dome and it slides down the oof. 0 m (a) (b) (c) (d) (e) Find an epession fo the velocit v of the object when its position vecto makes an angle θ with the vetical though the cente of the dome (see the figue). While it is in contact with the oof the object is undegoing motion in a cicula ac. Wite down an epession fo its centipetal acceleation as a function of angle θ. Find an epession fo the nomal contact foce as a function of angle θ and the mass m of the object. Hence detemine the angle θ at which the object leaves the oof, as well as its height and speed at this point. Find the distance fom the outside wall of the dome at which the object hits the gound. Mechanics Topic D (otation) - 5 David psle
. NGUL DYNMCS. Toque ais F The toque (o moment of foce) T about an ais is given b: toque foce pependicula distance fom ais T F (6) Toque measues the tuning effect of a foce. When the foce is not pependicula to the adius vecto then onl the component pependicula to the adius vecto contibutes toque. v. ngula Momentum ngula momentum (o moment of momentum) h is given b: angula momentum momentum pependicula distance fom ais h mv m ω ais m (7) (Fo non-cicula motion, v is the tansvese component of velocit see Section.4.). The ngula-momentum Pinciple Fo Motion in a Cicle Foce-momentum pinciple: F d ( mv ) f F is the tangential component of foce and is constant (i.e. cicula motion) then F d ( mv ) toque = ate of change of angula momentum (8) n fact, (8) holds fo non-cicula motion, but the poof equies moe effot; see Section.4. Equation (8) is the otational analogue of the momentum pinciple fo tanslational motion: foce = ate of change of momentum Fo single paticles the angula-momentum equation offes no advantage ove the momentum equation. Howeve, it is invaluable in the teatment of sstems of paticles and, in paticula, igid-bod otation, to which it ma be applied b summing ove all masses in the sstem. The toque is then the sum of the moments of the etenal foces onl, since the intenal foces between paticles ae equal and opposite and cancel in pais. Whilst one can have a moment of an phsical quantit, toque is used almost eclusivel fo moment of foce. Mechanics Topic D (otation) - 6 David psle
Eample 7. (Ohanian) The oiginal Feis wheel built b Geoge Feis had adius 8 m and mass.90 6 kg. ssuming that all of its mass was unifoml distibuted along the im of the wheel, if the wheel was initiall otating at 0.05 ev min, what constant toque would stop it in 0 s? What foce eeted on the im of the wheel would have given such a toque? n the absence of an etenal toque, a diect coolla of the angula-momentum pinciple is: The Pinciple of Consevation of ngula Momentum The angula momentum of an isolated sstem emains constant..4 The ngula-momentum Pinciple Fo bita Motion Fo a paticle of mass m, the angula momentum is h sin α mv mvsin α i.e. onl the tansvese component of velocit, v = v sin α, contibutes to the angula momentum. The adial component v = v cos α has no moment about the ais. simila statement applies fo the toque when the foce vecto is split into tansvese and adial components. O O sin P v sin P v v v cos Using a vecto coss poduct (denoted b ), both angula momentum and toque ma be epesented b vectos oiented along the otation ais (in the sense of a ight-hand scew): angula momentum: h mvsin α o h mv (9a) toque: T F sin α o T F (9b) Diffeentiating the vecto epession fo angula momentum, using the poduct ule: dh d d mv ( mv) v mv F 0 F Hence, dh T which is, in vecto fom, the angula-momentum pinciple: ate of change of angula momentum = toque. (0) B summing ove all paticles this can be applied to the whole sstem, with T the toque due to etenal foces onl. Mechanics Topic D (otation) - 7 David psle
. GD-BODY OTTON. Moment of netia Eample. biccle wheel and a flat disk have the same mass, the same adius and ae spinning at the same ate. Which has the geate angula momentum and kinetic eneg? Fo otating igid bodies, diffeent paticles lie at diffeent adii and hence have diffeent speeds. Paticles at geate adius move faste and contibute moe to the bod s angula momentum and kinetic eneg. Thus, the angula momentum and kinetic eneg depend on the distibution of mass elative to the ais of otation. The total angula momentum and kinetic eneg ma be obtained b summing ove individual paticles of mass m at adius. Most impotantl, although paticles at diffeent adii have diffeent speeds v, the all have the same angula velocit ω. ngula Momentum H mv ( m( ω) m ) ω m Kinetic Eneg K ( mv m( ω) m ) ω The quantit m () is the moment of inetia (o second moment of mass) of the bod about the specified ais. With this definition, angula momentum kinetic eneg (c.f. momentum P = Mv and kinetic eneg H ω () K ω () K Mv fo tanslation). Moment descibes the distibution of mass elative to the selected ais. (t gives highe weighting to masses at geate adii.) netia efes to a esistance to a change in motion (acceleation). n this sense, the moment of inetia fulfils the same ole fo otation as the mass of a bod in tanslation. Mechanics Topic D (otation) - 8 David psle
Eample evisited. Fo a hoop (a close appoimation to the biccle wheel) all the mass is concentated at the same adius. Hence m m M Fo a flat disk it tuns out (see late) that the moment of inetia is ½M. Othe things being equal, the disk will have half the angula momentum and half the kinetic eneg of the hoop. This is because some of its mass is at a smalle adius and is moving moe slowl.. Second Moments and the adius of Gation The moment (stictl, the fist moment) of an quantit is defined b fist moment = quantit distance Similal, second moment = quantit (distance) Fo an etended bod the distance vaies, so we must sum ove constituent pats; e.g. m second moment of mass (4) (n ou Hdaulics and Stuctues couses ou will come acoss an eactl analogous quantit the second moment of aea in connection with hdostatic foces on dams and esistance to bending of beams, espectivel.) The cente of mass is that point at which the same concentated mass would have the same fist moment: M m The adius of gation, k, is that adius at which the same mass would have the same second moment: M k m (5) Distibuted mass Concentated mass k Same total mass and moment of inetia Mechanics Topic D (otation) - 9 David psle
Eamples. Hoop of mass M and adius (ais though cente, pependicula to plane) Moment of inetia, M adius of gation, k = The adius of gation equals the geometic adius because all the mass is concentated at the cicumfeence. Cicula disc of mass M and adius (ais though cente, pependicula to plane) Moment of inetia M adius of gation k The adius of gation is less than the geometic adius because mass is distibuted ove a ange of distances fom the ais. Mechanics Topic D (otation) - 0 David psle
. The Equation of otational Motion Fo igid-bod otation the equation of motion is the angula momentum equation: d T ( ω) toque ate of change of angula momentum (6) This is the otational equivalent of Newton s Second Law: F d ( Mv ) foce = ate of change of momentum Fo solid bodies, since moment of inetia and mass ae usuall constant we usuall wite these in tems of acceleation: dω dv T (otation), F M (tanslation) (7) f we integate (6) with espect to time we obtain an impulse equation: B T ( ω) B ( ω) toque time change in angula momentum (8) The LHS is called the angula impulse. ltenativel we can integate (6) with espect to angle to obtain an eneg equation. Fist ewite it as d d( ω) dθ d( ω) d T ( ω) ω ( ω ) dθ dθ dθ ntegating with espect to angle θ gives the Mechanical Eneg Pinciple: B T dθ ( ω ) B ( ω ) wok done ( toque angle) change in kinetic eneg (9) Mechanics Topic D (otation) - David psle
.4 Compaing Tanslation and otation Tanslation otation Displacement θ Velocit v ω cceleation a α netia m Effective location of mass cente of mass adius of gation Cause of motion foce toque Tanslation otation Momentum mv ω Kinetic eneg mv ω Powe Fv Tω Equation of motion ate fom foce = ate of change of momentum F d ( mv ) toque = ate of change of angula momentum d T ( ω) Equation of motion impulse fom impulse (foce time) = change of momentum B F ( mv) B ( mv) angula impulse (toque time) = change of angula momentum B T ( ω) B ( ω) Equation of motion eneg fom wok done (foce distance) = change of kinetic eneg B F d ( mv ) B ( mv ) wok done (toque angle) = change of kinetic eneg B T dθ ( ω ) B ( ω ) Mechanics Topic D (otation) - David psle
.5 Eamples Eample 8. bucket of mass M is fastened to one end of a light inetensible ope. The ope is coiled ound a windlass in the fom of a cicula clinde (adius, moment of inetia ) which is left g fee to otate about its ais. Pove that the bucket descends with acceleation. M M Mg Eample 9. flwheel whose aial moment of inetia is 000 kg m otates with an angula velocit of 00 pm. Find the angula impulse which would be equied to bing the flwheel to est. Hence, find the fictional toque at the beaings if the flwheel comes to est in 0 min unde fiction alone. Eample 0. flwheel of adius 500 mm is attached to a shaft of adius 00 mm, the combined assembl having a moment of inetia of 500 kg m. Long cables ae wapped aound flwheel and shaft in opposite diections and ae attached to masses of 0 kg and 0 kg espectivel, which ae initiall at est as shown. Calculate: (a) how fa the 0 kg mass must dop in ode to aise the 0 kg mass b m; (b) the angula velocit of the shaft at this point. 500 mm 00 mm 0 kg 0 kg Mechanics Topic D (otation) - David psle
Eample. 5 kg mass hangs in the loop of a light inetensible cable, one end of the cable being fied and the othe wound ound a wheel of adius 0. m and moment of inetia 0.9 kg m so that the lengths of cable ae vetical (see the figue). The mass is eleased fom est and falls, tuning the wheel. Neglecting fiction between the mass and the loop of cable and between the wheel and its beaings, find: (a) a elationship between the downwad velocit of the mass, v, and the angula velocit of the wheel, ω; (b) the downwad acceleation of the mass; (c) the speed of the mass when it has fallen a distance m; (d) the numbe of tuns of the wheel befoe it eaches a otation ate of 00 pm. 5 kg Eample. squae plate of mass 6 kg and side 0. m is suspended veticall fom a fictionless hinge along one side. t is stuck dead cente b a lump of cla of mass kg which is moving at 0 m s hoizontall and emains stuck (totall inelastic collision). To what height will the bottom of the plate ise afte impact? (The moment of inetia of a squae lamina, side a and mass M, about one side, is Ma ) Mechanics Topic D (otation) - 4 David psle
4. CLCULTON OF MOMENTS OF NET 4. Methods of Calculation The moment of inetia depends on: the distibution of mass; the ais of otation. Some common methods of calculating ae as follows. Method. Fist Pinciples m Fo isolated paticles this can be done b diect summation. Fo continuous bodies integation is necessa. Method. Combination of Fundamental Elements (Hoop, Disk, od) hoop suface of evolution Fist pinciples disc solid of evolution od ectangula lamina Method. Stetching Paallel to the is f a shape is simpl stetched paallel to an ais then the moment of inetia is unchanged since the elative disposition of mass about the ais is not changed. e.g. hoop clindical shell disc solid clinde od ectangula lamina Method 4. Change of is hoop/disc od clinde ectangle Calculations ma be pefomed fist about some convenient (tpicall smmet) ais; the moment of inetia about the actual ais is then detemined b one of two techniques fo changing aes: the paallel-ais theoem and the pependicula-aes theoem. Mechanics Topic D (otation) - 5 David psle
4. Fundamental Shapes 4.. Hoop Fo a hoop (an infinitesimall-thin cicula ac) of mass M and adius, otating about its smmet ais, all the mass is concentated at the single distance fom the ais. Hence, Fo a hoop of mass M and adius, about the smmet ais pependicula to its plane: M (0) 4.. Disc Conside the moment of inetia of a unifom cicula disc (an infinitesimall-thin, cicula plane lamina) of mass M and adius, about the ais of smmet pependicula to its plane. The disc can be boken down into sub-elements which ae hoops of adius and thickness. Let ρ be the mass pe unit aea. m ρ (π δ) Sum ove all elements: π 4 m ρ π d ρ 0 M ρπ M Fo a disc of mass M and adius, about the smmet ais pependicula to its plane: M () 4.. od Conside the moment of inetia of a od (an infinitesimall-thin line segment) of mass M and length L, about its ais of smmet. The od can be boken down into sub-elements of length δ, distance fom the ais. Let ρ be the mass pe unit length. m ρ δ Summing: m ρ L/ L/ d ρl M ρl L M Fo a od of mass M and length L, about its ais of smmet: ML () L Mechanics Topic D (otation) - 6 David psle
4. Stetching Paallel to an is The distibution of mass about the ais and hence the moment of inetia is not changed b stetching paallel to the ais of otation without change of mass. Hence, fo the aes shown: hoop clindical shell: disc solid clinde: od ectangula lamina: M M Ma hoop/disc clinde (n the last case the ais is in the plane of the lamina.) b The onl dependence on the dimension paallel to the ais of otation is via its effect on the total mass M. od a ectangle Eample. oute steel im (pat (b)) 40 cm 6 cm flwheel shaft 0 cm (a) (b) (c) 0 cm flwheel consists of an aluminium disc of diamete 40 cm and thickness 6 cm, mounted on an aluminium shaft of diamete 0 cm and length 0 cm as shown. Calculate the moment of inetia of flwheel + shaft. To incease the moment of inetia a steel im is fied to the outside of the flwheel. Calculate the oute adius of the steel im equied to double the moment of inetia of the assembl. f the flwheel is initiall otating at 00 pm, calculate the constant fictional baking foce which needs to be applied to the outside of the steel im in pat (b) if the flwheel is to be bought to est in 0 seconds. Fo this question ou ma equie the following infomation. Densit of aluminium: 650 kg m ; steel: 7850 kg m. Moment of inetia of a solid clinde of adius and mass M about its ais: M. Mechanics Topic D (otation) - 7 David psle
Mechanics Topic D (otation) - 8 David psle 4.4 Volumes of evolution Moments of inetia fo volumes of evolution ma be deduced b summing ove infinitesimal discs (o ve thin clindes) of adius and length δ. Let ρ be the mass pe unit volume. Then the elemental mass and moment of inetia ae, espectivel: mass: ) δ ρ (π δ m moment of inetia: m adius disk mass δ π ρ δ δ 4 Summing ove all elemental masses and moments of inetia: m M d π ρ δ d π ρ δ 4 () Eample. Find the moment of inetia of a solid sphee, mass M and adius about an ais of smmet. Fo a solid sphee,. Hence, between. Thus, 4 5 5 8 4 4 π ρ π ρ )d ( π ρ d ) ( π ρ M Hence, 5 M whence 5 M ais
4.5 Change of is 4.5. Paallel-is ule f the moment of inetia of a bod of mass M about an ais though its cente of mass is G, then the moment of inetia about a paallel ais is given b G Md whee M is the mass of the bod and d is the distance between aes. Poof. Take (,,z) coodinates elative to the cente of mass, with the z diection paallel to the aes of otation. B Pthagoas, m m( ) G mp m[( ) ( ) )] G (0,0) d P(,) (, ) Epanding the second of these: m( m( ) m( G Md 0 0 The last two tems vanish because thee ae no moments about the cente of mass. ) ) m m Coolla. The coesponding adii of gation ae elated b k k d G Coolla. Fo a set of paallel aes, the smallest moment of inetia is about an ais though the cente of mass. Eample. The M.. fo a od of mass M and length L about an ais though its cente and nomal to the od is G ML. Hence the M.. about a paallel ais though the end of the od is G M ( L) ML ML 4 ML G L L Mechanics Topic D (otation) - 9 David psle
4.. Pependicula-is ule mpotant note. This is applicable to plane laminae onl. Howeve, it can often be combined with stetching paallel to the ais to give -d shapes. f a plane bod has moments of inetia and about pependicula aes O and O in the plane of the bod, then its moment of inetia about an ais Oz, pependicula to the plane, is: z Poof. B Pthagoas, Hence m z m m z Eample. Find the moment of inetia of a ectangula lamina, mass M and sides a and b, about an ais though the cente, pependicula to the lamina. Solution. Fom the ealie eamples, the moments of inetia about aes in the plane of the lamina ae, Ma Mb Hence, z M( a b ) b a z Eample. Find the moment of inetia of a cicula disc, adius, about a diamete. Solution. n this case we use the pependicula-ais theoem in evese, because we alead know the moment of inetia about an ais though the cente, pependicula to the plane of the disc: z M. B otational smmet the unknown moment of inetia about a diamete is the same fo both and aes. Hence, M whence M 4 M z Mechanics Topic D (otation) - 0 David psle
Eample 4. Find the adius of gation of the squae-fame lamina shown about an ais along one side. ais ais 0. m 0.5 m 0. m 0.5 m Eample 5. igid famewok consists of fou ods, each of length L and mass M, connected in the fom of a squae BCD as shown. Find epessions, in tems of L and M, fo the moment of inetia of the famewok about: (a) the ais of smmet SS; (b) the side B; (c) an ais pependicula to the famewok and passing though cente O; (d) an ais pependicula to the famewok and passing though vete ; (e) the diagonal C. Data: the moment of inetia of a od, length L and mass M, about an ais though its cente and pependicula to the od is ML. B S C O S D Mechanics Topic D (otation) - David psle
5. GENEL MOTON OF GD BODY (Optional) The motion of a igid bod which is allowed to otate as well as tanslate (e.g. a olling bod) can be decomposed into: motion of the cente of mass otation of the bod unde the esultant etenal f oce elative to the cente of mass t ma be shown (optional eecise) that, fo a sstem of paticles (e.g. a igid bod): () The cente of mass moves like a single paticle of mass M unde the esultant of the etenal foces: dv d F M whee V d t () The elationship toque = ate of change of angula momentum : dh T holds fo the toque of all etenal foces about a point which is eithe: fied; o moving with the cente of mass. () The total kinetic eneg can be witten as the sum: kinetic eneg of the cente of mass ( MV ) + kinetic eneg of motion elative to the cente of mass Fo a igid bod, motion elative to the cente of mass must be otation and hence: K MV Gω tanslational KE otational KE about total kinetic eneg of cente of mass cente of mass 5. olling Without Slipping Conside a bod with cicula coss-section olling along a plane suface. f the bod olls without slipping then the distance moved b the point of contact must equal the length of ac swept out: s θ v Hence the linea and angula velocities ae elated b: dθ v ω The instantaneous point of contact with the plane has zeo velocit; hence the fiction foce does no wok but it is esponsible fo otating the bod! Mechanics Topic D (otation) - David psle
The total kinetic eneg is given b K ( tanslational KE) ( otational KE) mv ( m ω )ω Eample 6. (Snge and Giffiths) wheel consists of a thin im of mass M and n spokes each of mass m, which ma be consideed as thin ods teminating at the cente of the wheel. f the wheel is olling with linea velocit v, epess its kinetic eneg in tems of M, m, n, v. common eample is of a spheical o clindical bod olling down an inclined plane. The foces on the bod ae its weight Mg, the nomal eaction foce and the fiction foce F. v mg F Conside the linea motion of the cente of mass and the otational motion about it. foce = mass acceleation fo tanslation of the cente of mass: dv Mg sin θ F M (along slope) Mg cos θ 0 (nomal to slope) toque = ate of change of angula momentum fo otation about the cente of mass: dω F v and ω ae elated, if not slipping, b v = ω. Mechanics Topic D (otation) - David psle
Eample. (Ohanian) piece of steel pipe, mass 60 kg, olls down a amp inclined at 0 to the hoizontal. What is the acceleation if the pipe olls without slipping? What is the magnitude of the fiction foce that acts at the point of contact between the pipe and amp? Solution. Linea motion: Mg sin θ F Ma (i) otation about cente of mass: F α (ii) Eliminate F b (i) + (ii), noting that α a / : Mg sin θ ( M ) a Hence, Mg sin θ g sin θ a M M But fo a hoop, and hence (b stetching paallel to the ais) a pipe, a g sin θ 9.8 sin 0.45 m s This is the linea acceleation. M. Thus, Fo the fiction foce use eithe linea o otational equation of motion; e.g. fom (i): F Mg sin θ Ma M ( g sin θ a) 60 (9.8 sin 0.45) 88.7 N nswe:.45 m s ; 88 N. Mechanics Topic D (otation) - 4 David psle
5. olling With Slipping Fo a bod which is olling along a suface the condition fo no slipping is that the instantaneous point of contact is not moving; that is, the linea velocit of the cente of mass (v) must be equal and opposite to that of the elative velocit of a point on the im which is otating (ω). Hence, slipping occus whilst v ω. v n this case, fiction will act to oppose slipping. f a spinning bod is placed on a suface then it is the fiction foce which initiates its fowad motion. Note that, while slipping occus, thee is elative motion and so fiction is maimal: F μ F Eample. (Snge and Giffiths) hollow spheical ball of adius 5 cm is set spinning about a hoizontal ais with an angula velocit of 0 ad s. t is then gentl placed on a hoizontal plane and eleased. f the coefficient of fiction between the ball and the plane is 0.4, find the distance tavesed b the ball befoe slipping ceases. [The moment of inetia of a spheical shell of mass m and adius is m ]. Solution. nitiall slipping must occu, because the ball is not moving fowad but it is otating. Whilst slipping it is fiction which (a) acceleates the tanslational motion fom est; (b) deceleates the otation. Slipping ceases when v = ω, but until this point fiction is maimal and given b F μ μmg. Linea motion dv F m Whilst slipping, F μ μmg. Hence, dv μmg m d v μg with v = 0 at t = 0. v μgt (i) mg otational motion dω F Whilst slipping, F μ μmg. lso, m. Hence, Mechanics Topic D (otation) - 5 David psle
dω μmg m dω μg with ω = ω 0 = 0 ad s at t = 0. ω μgt (ii) ω 0 Slipping stops when v = ω. Fom (i) and (ii), this occus when gt ω μgt μ 0 5 μgt ω 0 ω0 t 5 μg This distance tavelled ma be detemined fom the linea constant-acceleation fomula s ut at, with ω0 u = 0, a = μg, t 5 μg Hence, ω0 ω0 s 0 μg ( ) 5 μg 5 μg Using consistent length units of metes: 0.05 0 s 6.000 m 5 0.4 9.8 nswe: 6.0 mm. Mechanics Topic D (otation) - 6 David psle
ppendi : Moments of netia fo Simple Shapes Man fomulae ae given in the tetbooks of Meiam and Kaige o Gee and Timoshenko. Onl some of the moe common ones ae summaised hee. Geometic figues ae assumed to have a unifom densit and have a total mass M. Geomet is od, length L () Though cente ML ectangula lamina, sides L (pependicula to ais) and W () End ML () n-plane; smmet ML () Side ML () Pependicula to plane; smmet M ( L W ) Tiangula lamina, base B, altitude H Base 6 MH Cicula ing, adius () Pependicula to plane; smmet M () Diamete M Cicula disc, adius () Pependicula to plane; smmet M () Diamete 4 M Cicula clinde, adius, height H. Smmet ais M Solid sphee (o hemisphee), adius n diamete 5 M Spheical (o hemispheical) shell, adius n diamete M Moments of inetia fo man diffeent shapes o aes can be constucted fom these b: use of the paallel-ais o pependicula-aes ules; stetching paallel to an ais (without change of mass distibution); combination of fundamental elements. ppendi : Second Moment of ea Second moment of aea athe than second moment of mass appeas in stuctual engineeing (esistance to beam bending) and hdostatics (pessue foce). The fomulae fo second moments of aea of plane figues ae the same as those in the table above ecept that mass M is eplaced b aea. The same smbol () is used. Dependence on a length dimension paallel to the ais is often hidden inside M o ; e.g. second moment of aea of a ectangula lamina about an in-plane smmet ais: WL L (since WL ) second moment of aea of a tiangula lamina about a side of length B: BH H (since BH ) 6 You will meet second moments of aea a geat deal in ou Stuctues couses. Mechanics Topic D (otation) - 7 David psle
Numeical nswes to Eamples in the Tet Full woked answes ae given in a sepaate document online. Eample..750 ad s Eample..8 Eample. 0.9 Eample 4. (a) 0.680; (b) 0.0746 Eample 5. (a) 4 N; (b).6 m; (c) 9.88 m s Eample 6. v (a) v g( cosθ) 9.4( cosθ) ; (b) g( cosθ) 9.6( cosθ) (c) mg( cosθ ) ; (d) 48. ;. m;.4 m s ; (e).49 m Eample 7. 4.790 5 N m; 600 N Eample 8. No numeical answe Eample 9..40 4 N m s; 5.4 N m Eample 0. (a) 5 m; (b).08 ad s Eample. (a) ω v ; (b).68 m s ; (c).7 m s ; (d) 4.4 evs Eample. 0.6 m Eample. (a) 0.407 kg m ; (b) 5 mm; (c). N Eample 4. 0.0 m Eample 5. (a) ML ; (b) 5 ML ; (c) 4 ML ; (d) 0 ML ; (b) ML Mechanics Topic D (otation) - 8 David psle
Eample 6. ( M nm ) v Mechanics Topic D (otation) - 9 David psle