Chapter Torque and Angular Momentum I. Torque II. Angular momentum - Defnton III. Newton s second law n angular form IV. Angular momentum - System of partcles - Rgd body - Conservaton I. Torque - Vector quantty. τ = r F Drecton: rght hand rule. Magntude: τ = r F snϕ = r F = ( r snϕ) F = r F Torque s calculated wth respect to (about) a pont. Changng the pont can change the torque s magntude and drecton.
II. Angular momentum - Vector quantty. l = r p = m( r v) Unts: kg m 2 /s Magntude: l = r psnϕ = r m vsnϕ = r m v = r p = ( r snϕ) p = r p = r m v Drecton: rght hand rule. l postve counterclockwse l negatve clockwse Drecton of l s always perpendcular to plane formed by r and p. III. Newton s second law n angular form Lnear dp F net = Angular dl τ net = Sngle partcle The vector sum of all torques actng on a partcle s equal to the tme rate of change of the angular momentum of that partcle. Proof: dl dv dr l = m( r v) = m r + v = m r dl = r ma = r Fnet = ( r F ) = τ net ( a + v v) = m( r a) = V. Angular momentum n L = l + l + l + + l = - System of partcles: 2 3... n l = 2
dl n dl = = n = τ = net, τ net dl = Includes nternal torques (due to forces between partcles wthn system) and external torques (due to forces on the partcles from bodes outsde system). Forces nsde system thrd law force pars torque nt sum =0 The only torques that can change the angular momentum of a system are the external torques actng on a system. The net external torque actng on a system of partcles s equal to the tme rate of change of the system s total angular momentum L. - Rgd body (rotatng about a fxed axs wth constant angular speed ω): Magntude v = ω r l = r )( p )(sn 90 ) = ( r )( m v ) ( l = r m 2 ( ω r ) = ω mr Drecton: l perpendcular to r and p n n n L = 2 z lz = mr ω = 2 m r ω = Iω = = = L = ω I z L = Iω dlz dω dlz = I = Iα = τ ext Rotatonal nerta of a rgd body about a fxed axs 3
- Conservaton of angular momentum: Newton s second law dl τ net = If no net external torque acts on the system (solated system) dl = 0 L = cte Law of conservaton of angular momentum: L = L ( solated system) f Net angular momentum at tme t = Net angular momentum at later tme t f If the net external torque actng on a system s zero, the angular momentum of the system remans constant, no matter what changes take place wthn the system. If the component of the net external torque on a system along a certan axs s zero, the component of the angular momentum of the system along that axs cannot change, no matter what changes take place wthn the system. Ths conservaton law holds not only wthn the frame of Newton s mechancs but also for relatvstc partcles (speeds close to lght) and subatomc partcles. I ω = I ω f f ( I,f, ω,f refer to rotatonal nerta and angular speed before and after the redstrbuton of mass about the rotatonal axs ). 4
Examples: Spnnng volunteer I f < I (mass closer to rotaton axs) Torque ext =0 I ω = I f ω f ω f > ω Sprngboard dver - Center of mass follows parabolc path. - When n ar, no net external torque about COM Dver s angular momentum L constant throughout dve (magntude and drecton). - L s perpendcular to the plane of the fgure (nward). - Begnnng of dve She pulls arms/legs closer Intenton: I s reduced ωncreases - End of dve layout poston Purpose: I ncreases slow rotaton rate less water-splash 5
Force Translaton F Torque Rotaton τ = r F Lnear momentum p Angular momentum l = r p Lnear momentum (system of partcles, rgd body) Newton s second law Conservaton law P = p = Mv dp F = (Closed solated system) COM P = cte Angular momentum L Newton s second law = l L = Iω τ net Conservaton law System of partcles Rgd body, fxed axs L=component along that axs. dl = (Closed solated system) L = cte IV. Precesson of a gyroscope Gyroscope: wheel fxed to shaft and free to spn about shaft s axs. Non-spnnng gyroscope dl τ = If one end of shaft s placed on a support and released Gyroscope falls by rotatng downward about the tp of the support. The torque causng the downward rotaton (fall) changes angular momentum of gyroscope. Torque caused by gravtatonal force actng on COM. τ = Mgr sn 90 = Mgr 6
Rapdly spnnng gyroscope If released wth shaft s angle slghtly upward frst rotates downward, then spns horzontally about vertcal axs z precesson due to non-zero ntal angular momentum Smplfcaton: ) L due to rapd spn >> L due to precesson ) shaft horzontal when precesson starts L = Iω I = rotatonal moment of gyroscope about shaft ω = angular speed of wheel about shaft d L = τ dl = τ = Mgr Vector L along shaft, parallel to r Torque perpendcular to L can only change the Drecton of L, not ts magntude. dl Mgr d ϕ = = L Iω Rapdly spnnng gyroscope d L = τ dl = τ = Mgr dl Mgr d ϕ = = L Iω Precesson rate: dϕ Mgr Ω = = Iω 7