PHYS 172H: Moden Mechanics Fall 2010 Lectue 19 ngula momentum Chapte 11.4 11.7
The angula momentum pinciple dp = F dl =? net d ( p ) d dp = p+ = v γ mv = = 0 The angula momentum pinciple fo a point paticle dl = Fnet = τ Δ L = F Δt ( ) net = τ Δt F net p toque : τ F F net net Note: The angula momentum pinciple can be deived fom the momentum pinciple, but is ultimately, independently, equally basic and fundamental.
Clicke Toque at angle toque : τ F net τ = F net sinθ B τ C τ = F net 1. Which case coesponds to lagest toque? 2. The diection of Toque in case B is: ) into page B) out of page C) it has no diection since it is zeo
Consevation of angula momentum Δ L +Δ =, system L, suoundings 0 Impotant: both L s must be about the same point (axis) Example: Hee take system to be (peson + platfom) ; suoundings don t change This example is essentially the same as the opening slides of Lectue 18, the ed ball stiking and sticking to the geen wheel which then spins.
comet B F gav dl = F =τ net CLICKER: What is the diection of the toque on the comet at point B about the sta due to gavitational pull? ) Into the page B) Out of the page C) It is zeo L = p mv mv 1 1 = 2 2 v 1 1= v 2 2 (nonelativistic)
Example: Keple and elliptical obits Keple, 1609: a adius vecto joining any planet to the Sun sweeps out equal aeas in equal lengths of time Can be easily poven using consevation of angula momentum See book p. 430 (11.4)
Clicke ball falls staight down in the xy plane. Its momentum is shown by the blue aow. What is the diection of the ball's angula momentum about the oigin? ) +y y B) y C) +z (out of the page) D) z (into the page) E) zeo magnitude x L = p
Clicke L = p p planet obits a sta, in a cicula obit in the xy plane. Its momentum is shown by the ed aow. What is the diection of the angula momentum of the planet with espect to the sta? ) same diection as p B) opposite to p C) into the page D) out of the page E) zeo magnitude
dl dl2 dl3 1 = F + f + f 1 1, ext 1 12 1 13 = F + f + f 2 2, ext 2 21 2 23 = F + f + f 3 3, ext 3 31 3 32 Multipaticle system 1 2 F 1,ext Net toque caused by intenal foces cancels out! = f 12 2 f 13 m 1 f m 2 23 1 F 2,ext f 21 f f 32 31 m 3 3 F 3,ext L tot, τ net, ext, ( ) d L1+ L2 + L3 dl1 dl2 dl3 = + + = F + F + F 1 1, ext 2 2, ext 3 3, ext
The angula momentum pinciple fo a multipaticle system dltot, = τ net, ext, Δ L = τ Δt tot, net, ext, The angula momentum pinciple elative to the cente of mass: Zeo length! dlcm d ( ) dl = cm, cm Ptot + L ot = dlot = τ net, cm Δ L = τ Δt ot net, cm ot
The thee pinciples of mechanics Momentum ngula momentum Enegy (fundamental) dp = F net Extenal foce: momentum changes (deived fo point mass) dl =τ Δ E = W + Q Extenal toque: angula momentum changes (fundamental) Enegy input o output: enegy of the system changes No extenal foce: momentum is constant No extenal toque: angula momentum is constant No enegy in/output: enegy of the system is constant Location of object does not matte Location of object elative to does matte Location in time of object does not matte
The thee pinciples of mechanics Momentum ngula momentum Enegy (fundamental) dp = F net (fundamental) dl =τ Δ E = W + Q (fundamental) No extenal foce: momentum is constant (conseved) No extenal toque: angula momentum is conseved No enegy in/output: enegy of the system is conseved Location of system does not matte TRNSLTIONL INVRINCE OF THE LWS OF PHYSICS ngula oientation of system about its CM does not matte. ROTTIONL INVRINCE OF LWS Ea of the system does not matte TIME INVRINCE OF THE LWS OF PHYSICS
ngula momentum: a system with no toque Doothy Hamill, 1985 dlot = τ net, cm = 0 Ii > I f L = Iω I = m 2 m 2 + + ot 1 1 2 2... I ω = Iω Ii ω f = ωi I f f i i f http://www.hep.phys.soton.ac.uk/couses/phys2006/
Consevation of angula momentum In a closed system L = Iω If I is changed then ω will change (like a skate). We can also change one pat of a system fo example if we invet the bicycle wheel I does not change but ω will change fo the est of the system to keep L constant What happens if the gil moves to the oute edge of the platfom? 11/3/2010 14
ngula momentum: a system with no toque Cat always lands on its feet http://www.youtube.com/watch?v=rhhxbohk_hs
ngula momentum: application fee-falling cat cannot alte its total angula momentum. Nonetheless, by swinging its tail and twisting its body to alte its moment of inetia, the cat can manage to alte its oientation See also book example: High dive page 437
ngula momentum: application Space station and ed flywheel, neithe of them ae otating. L sys =0 Spin the flywheel clockwise, the space station has to spin counteclockwise to maintain total L = 0 Late, stop the flywheel spinning. Space station will stop spinning too. But it will have otated elative to the beginning of the whole sequence of opeations.
The GYROSCOPE Right-hand ule gets used hee, big-time! Cul finges in diection of otation (o of Toque). Thumb points along vecto. Note: Toque x (delta t) = ngula Impulse, which is exactly the CHNGE in angula momentum L. L of GYROSCOPE. T mg The weight foce, with the specified pivot point, causes a Toque vecto pointing at us, out of the plane of the figue. Toque causes CHNGE of ngula momentum, L, which DDS to EXISTING L of the gyoscope. This is a VECTOR addition!! X RESULT: xis of gyoscope pecesses towads us, at this instant. 11/3/2010 18
ngula momentum: application meteo ips though a satellite with sola panels. y Calculate: v x,v y of cente of mass ω f angula velocity x Momentum pinciple: ( Mv + mv θ) mv θ = ( Mv + mv θ) ( Mv + mv θ) cos, sin, 0 cos, sin, 0 1 1 x 2 y 2 m vx v v1 v2 cos M = + ( ) θ v = ( v v ) y m M 1 2 sinθ
ngula momentum: application meteo ips though a satellite with sola panels. Calculate: v x,v y of cente of mass ω f angula velocity ngula momentum pinciple: Iω + mvhcosθ = Iω + mv hcosθ i 1 f 2 Fo sphee: I = 2 5 MR 2 hm ω f = ωi + ( v 1 v 2) cosθ I Diection?
Static equilibium: seesaw dl = τ net, ext, = 0 τ net ext = ( M g) + ( M g) = ( M+ M) g= M = M,, 1 1 2 2 0 11 2 2 0 11 2 2 M d,0,0 = M d,0,0 1 1 2 2 M d = M d 1 1 2 2 1 2 cm = Total mass, acting at CM, exets zeo toque about the fulcum M M + M + M 11 2 2 1 2 = 0