# rotation -- Conservation of mechanical energy for rotation -- Angular momentum -- Conservation of angular momentum

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1 Final Exam Duing class (1-3:55 pm) on 6/7, Mon Room: 41 FMH (classoom) Bing scientific calculatos No smat phone calculatos l ae allowed. Exam coves eveything leaned in this couse. Review session: Thusday duing class 1 Rotational Motion -- Toque -- Kinetic enegy, potential enegy, mechanical enegy fo otation -- Consevation of mechanical enegy fo otation -- Angula momentum -- Consevation of angula momentum 1

2 How to find toque F θ Fom figue, F T m τ F F sinθ F T p axis p Line of action If we know components ( x, y,0) τ (0,0, F F) x y y x F ( F, F,0) x y Reminde: Rotational vaiables ae vectos, having diection The angula displacement, speed, and acceleation ( θ, ω, α ) ae vectos with diection. The diections ae given by the ight-hand ule: Finges of ight hand cul along the angula diection (See Fig.) Then, the diection of thumb is the diection i of the angula quantity.

3 iclicke quiz All foces have the same magnitude. Which foce has the geatest magnitude of toque with espect to the pivot? Pivot A B (a) A (b) B (c) C (d) All foces have the same toque (e) Not enough infomation C (1 m, m) 5.1. A paticle located at the position vecto has a foce F ( N,3 N) acting on it. The toque in N.m about the oigin is? 3

4 Gavitational foce on extended objects Toque on extended object by gavitational foce Assume that the total gavitational foce effectively acts at the cente of mass. Gavitational potential enegy of extended object M g H, whee H is the height of the cente of mass and M is the total mass. iclicke Q Axis of otation Hoizontal unifom od of length L & mass M Find the toque by gavitational foce. A. LMg B. (L/)Mg C. LMg D. (3/)LMg E. None of the above Find the angula acceleation. I end, od 1 3 ML 4

5 Example: Toque and Angula Acceleation of a Wheel Cod wapped aound disk, hanging weight Cod does not slip o stetch Let m 1. kg, M.5 kg, 0. m Find acceleation of mass m, find angula acceleation α fo disk, tension, and toque on the disk a Fomula: I 1 M mg Example: A unifom cicula disk of adius and mass M is pulled by constant hoizontal foce F applied to the cente of mass, and is olling without slipping. M kg, 0.5 m, I_(cm,disk)(1/)MR^, F5N. a) Find the angula acceleation. b) Find the static fiction foce. CCW + cm P F f s iclicke: If thee s no fiction fom the suface on the disk, and if the disk is initially not otating, would it otate about cente of mass by applied foce? A) Yes B) No 5

6 Kinetic enegy fo otation If the igid body otates with espect to an axis fixed in space 1 K Iω Kinetic enegy fo otation If the igid body otates with espect to a moving axis though the cente of mass, such as olling ω K K + K K ot K cm ot cm 1 Icmω 1 Mv cm v cm Fo olling without slipping Δ s RΔθ vcm Rω 6

7 Fo any otational o tanslational motion, U gavity Mgh cm E K + U mech Δ E W mech g non consevative If W 0 non consevative E, then is conseved. mech Example of enegy consevation Axis of otation Initial Final Hoizontal unifom od of length L & mass M is eleased fom est. Find its angula speed at the lowest point, assuming no fiction between axis of otation and the od. 7

8 Example: Use enegy consevation to find the speed of the bowling ball as it olls w/o slipping to the bottom of the amp Hint: Given: hm Fomula: Fo a solid sphee Icm 5 MR Rotation acceleates if thee is fiction between the sphee and the amp Fiction foce poduces the net toque and angula acceleation. Thee is no mechanical enegy change because the contact point is always at est elative to the suface, so no wok is done against fiction iclicke Q: A solid sphee and a spheical shell of the same adius and same mass M oll to the bottom of a amp without slipping fom the same height h. Tue o false? : The two have the same speed at the bottom. A) Tue B) False. Shell is faste. C) False. Solid sphee is faste. D) Not enough infomation. I_(cm, spheical shell) (/3) MR^ I_(cm, solid sphee)(/5) MR^ 8

9 Angula momentum concepts & definition - Linea momentum: p mv - Angula (Rotational) momentum: L moment of inetia x angula velocity Iω linea otational inetia speed linea momentum m v pmv I ω LIω igid body angula momentum Angula momentum of a bowling ball 6.1. A bowling ball is otating as shown about its mass cente axis. Find it s angula momentum about that axis, in kg.m /s A) 4 B) ½ C) 7 D) E) ¼ ω 4 ad/s M 5 kg ½ m I /5 MR L Iω 9

10 Angula momentum of a point paticle L Iω m ω mvt mv sin θ mvp pt pp O p vt θ vt v ω Note: L 0 if v is paallel to (adially in o out) L I ω mvt mv sin θ mvp p p sin θ p T p Angula momentum of a point paticle O p p T θ p L p psin θ p T p If we know components ( x, y,0) x L (0,0, p p) x y y x p ( p, p,0) y 10

11 Angula momentum fo ca 5.. A ca of mass 1000 kg moves with a speed of 50 m/s on a cicula tack of adius 100 m. What is the magnitude of its angula momentum (in kg m /s) elative to the cente of the ace tack (point P )? A) A B) C) D) E) P B 5.3. What would the angula momentum about point P be if the ca leaves the tack at A and ends up at point B with the same velocity? A) Same as above B) Diffeent fom above C) Not Enough Infomation L p p p sin( θ) Net angula momentum of paticles L L + L + L + net

12 Example: calculating angula momentum fo paticles PP1060-3*: Two objects ae moving as shown in the figue. What is thei total angula momentum about point O? m m 1 τ net Δω Iα I Δt ΔL Δt Δ L τ Δt net Consevation of angula momentum : Angula momentum, L is conseved if τ net 0 Consevation of net angula momentum fo a system Net angula momentum, L net is conseved if τ net,ext 0 1

13 A puck on a fictionless ai hockey table has a mass of 5.0 g and is attached to a cod passing though a hole in the suface as in the figue. The puck is evolving at a distance.0 m fom the hole with an angula velocity of 3.0 ad/s. The cod is then pulled fom below, shotening the adius to 1.0 m. The new angula velocity (in ad/s) is. Demonstation: Spinning Pofesso: Web link Isolated System τ net, ext 0 L constant I ω I ω i i f f Moment of inetia changes Anothe link 13

14 Intenal toques do not change total angula momentum it is edistibuted within the isolated system τ net 0 about z - axis L sys constant Demonstation: Weblink, moe Spacecaft maneuves by Intenal toques not 0 spinning the flywheel they evese L wh the caft counte-otates t t L boy, final L 0 L + L Lwheel tot caft flywheel Tetheed Astonauts 6.3. Two astonauts each having mass M ae connected by a mass-less ope of length d. They ae isolated in space, obiting thei cente of mass at identical speeds v. By pulling on the ope, one of them shotens the distance between them to d/. What ae the new angula momentum L and speed v? A) L mvd/ mvd/, v v v/ B) L mvd, v v C) L mvd, v v D) L mv d, v v/ E) L mvd, v v/ L Iω L mv T 14

15 Example: A mey-go-ound poblem A 40-kg child unning at 4.0 m/s jumps tangentially onto a stationay cicula mey-go-ound platfom whose adius is.0 m and whose moment of inetia is 0 kg-m. Find the angula velocity of the platfom afte the child has jumped on. 15

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