Angular Momentum and Fixed Axis Rotation. 8.01t Nov 10, 2004

Transcription

1 Angular Momentum and Fxed Axs Rotaton 8.01t Nov 10, 2004

2 Dynamcs: Translatonal and Rotatonal Moton Translatonal Dynamcs Total Force Torque Angular Momentum about Dynamcs of Rotaton F ext Momentum of a ystem p Dynamcs of Translaton dp Fext = Rotatonal Dynamcs of pont mass about τ = r, F L = r, m p d τ = L m m

3 Angular Velocty Vector and Angular Acceleraton Vector for Fxed Axs Rotaton Fxed axs of rotaton: z-axs Angular velocty vector ω = dθ kˆ Angular acceleraton vector α = 2 d θ kˆ 2

4 Angular Momentum of a Pont Partcle pont partcle of mass m movng wth a velocty v momentum p= mv Fx a pont vector r m, from the pont to the locaton of the object angular momentum about the pont L = r p, m

5 Cross Product: Angular Momentum of a Pont Partcle Magntude: L = r, p sn a) moment arm r = r L m θ, snθ m = r p b) Perpendcular momentum p = p L = r snθ p, T L = r p, m

6 Cross Product: Angular Momentum of a Pont Partcle Drecton Rght Hand Rule

7 Angular Momentum for Fxed Axs Rotaton Fxed axs of rotaton: z- axs Angular velocty v = r ω =rωkˆ angular momentum about the pont L = r p = r v, m m z-component of the angular momentum about, ˆ ˆ 2 L ˆ = r m m v= rmvk = rmrωk = mr ωk,

8 Fxed Axs Rotaton Angular Momentum about z-axs 2 Lz, = mr ω = I Rotatonal Dynamcs τ dl ω z, dω z, = = I = I α

9 PR Queston A person spns a tenns ball on a strng n a horzontal crcle (so that the axs of rotaton s vertcal). At the pont ndcated below, the ball s gven a sharp blow n the forward drecton. Ths causes a change n angular momentum dl n the 1. x drecton 2. y drecton 3. z drecton

10 PR Queston A dumbbell s rotatng about ts center as shown. Compared to the dumbbell's angular momentum about ts center, ts angular momentum about pont B s 1. bgger. 2. the same. 3. smaller.

11 Tme Dervatve of Angular Momentum for a Pont Partcle Tme dervatve of the angular momentum about : dl d = ( r ) m, p Product rule dl d dr ( ) m, d = rm, p = p+ rm, p Key Fact: = drm, d m, m m r = = v v v v 0 Result: dl d = rm, p= r m, F= τ

12 Torque and the Tme Dervatve of Angular Momentum for a Pont Partcle Torque about a pont s equal to the tme dervatve of the angular momentum about. τ d = L

13 Angular Momentum for a ystem Treat each partcle separately L = r p, m, Total Angular Momentum for ystem about L L r p = N = N =, =, m = 1 = 1 of Partcles

14 Angular Momentum and Torque for a ystem of Partcles Total torque about s the tme dervatve of angular momentum about dl dl = N = N = N, = = rm, F = τ, = τ = 1 = 1 = 1

15 Angular Momentum of a Rgd Body for Fxed Axs Rotaton Fxed axs of rotaton: z-axs angular momentum about the pont L = r p = r m v,,, z-component of the angular momentum about, L = r m v ( ), O, z r = r + r, O, O,

16 Z-component of the Angular Momentum about Mass element radus of the crcle momentum m r, mv z-component of the angular momentum about ( ) L = r mv, z, Velocty v = r ω, ummary: ( ) = = ( ) 2 L r mv m r ω, z,,

17 Z-component of the Angular Momentum about um over all mass elements Contnuous body Moment of Inerta Man Result ( ) ( ) ( ) 2 =, =, L L m r ω z z ( ) ( ) 2 L = dm r ω z I body, = dm( r ) z body ( ) =, L I ω z z 2

18 Torque and Angular Momentum for Fxed Axs Rotaton torque about s equal to the tme dervatve of the angular momentum about d = L resolved n the z-drecton ( τ ) τ ( ) ( ) 2 d L d I ω dω d θ z z, = = = I, z = I z, z = I 2, z α

19 Conservaton of Angular Momentum about a Pont Rotatonal dynamcs No external torques τ d = L d = = L 0 τ Change n Angular momentum s zero L L L = 0 ( ) ( ) 0 f Angular Momentum s conserved ( ) ( L ) = L f 0

20 PR Queston A fgure skater stands on one spot on the ce (assumed frctonless) and spns around wth her arms extended. When she pulls n her arms, she reduces her rotatonal nerta and her angular speed ncreases so that her angular momentum s conserved. Compared to her ntal rotatonal knetc energy, her rotatonal knetc energy after she has pulled n her arms must be 1. the same. 2. larger because she's rotatng faster. 3. smaller because her rotatonal nerta s smaller.

21 Conservaton Prncples Change n mechancal energy No non-conservatve work Change n momentum nc No external forces 0 W = E = K + U nc mechancal = W = E = K + U mechancal F external = = N d p = 1 0 = x = 0 = F = d d d p ( ) Fexternal ( p ) ( ) external ( p ) y x y

22 PR Queston A streetcar s freely coastng (no frcton) around a large crcular track. It s then swtched to a small crcular track. When coastng on the smaller crcle the streetcar's 1. mechancal energy s conserved and angular momentum about the center s conserved 2. mechancal energy s not conserved and angular momentum about the center s conserved 3. mechancal energy s not conserved and angular momentum about the center s not conserved 4. mechancal energy s conserved and angular momentum about the center s not conserved.

23 Total Angular Momentum about Total for translaton and rotaton about pont Orbtal angular momentum a Fxed Pont L = r m v + L spn, cm T cm cm L = r p orbtal, cm pn Angular Momentum for fxed axs rotaton L = I ω spn cm cm spn

24 Class Problem A meteor of mass m s approachng earth as shown on the sketch. The radus of the earth s R. The mass of the earth s me. uppose the meteor has an ntal speed of ve. Assume that the meteor started very far away from the earth. uppose the meteor just grazes the earth. The ntal moment arm of the meteor ( h on the sketch) s called the mpact parameter. The effectve scatterng angle for the meteor s the area. Ths s the effectve target sze of the earth as ntally seen by the meteor. π h 2

25 Class Problem a) Draw a force dagram for the forces actng on the meteor. b) Can you fnd a pont about whch the gravtatonal torque of the earth s force on the meteor s zero for the entre orbt of the meteor? c) What s the ntal angular momentum and fnal angular momentum (when t just grazes the earth) of the meteor? d) Apply conservaton of angular momentum to fnd a relatonshp between the meteor s fnal velocty and the mpact parameter. e) Apply conservaton of energy to fnd a relatonshp between the fnal velocty of the meteor and the ntal velocty of the meteor. f) Use your results n parts d) and e) to calculate the mpact parameter and the effectve scatterng cross secton.