Conservaton o Angular Momentum 8.0 WD Rotatonal and Translatonal Comparson Quantty Momentum Ang Momentum Force Torque Knetc Energy Work Power Rotaton L cm = I cm ω = dl / cm cm K = (/ ) rot P rot θ W = d θ θ0 = ω Icmω K = L /I rot cm cm Translaton K trans p = mv cm F p A ext total = d / = m cm = (/ ) mv K p m trans = / W = F d r 0 P = F v cm Torque and the Tme Dervatve o Angular Momentum Torque about a pont s equal to the tme dervatve o the angular momentum about. d = L Angular Momentum or a ystem o Partcles Treat each partcle separately L = r p,, Angular momentum or tem about L L r p = N = N =, =, = =
Angular Momentum and Torque or a ystem o Partcles Change n total angular momentum about a pont equals the total torque about the pont N N d L = = d, d =, = r +, p L p r = = = N = N = N dl d = p, =, = r r F, = = = = total d L ( ) = total Internal and External Torques The total external torque s the sum o the torques due to the net external orce actng on each element = N = N ext ext ext =, = r, F = = The total nternal torque arse rom the torques due to the nternal orces actng between pars o elements N = N = N = N = N nt nt nt =, j =,, j= r, F, j j= j= = = j= j j The total torque about s the sum o the external torques and the nternal torques total ext nt = + Internal Torques We know by Newton s Thrd Law that the nternal orces cancel n pars and hence the sum o the nternal orces s zero = N = N F, j= F 0= F, j j, = j= j Does the same statement hold about pars o nternal torques? nt nt,, j+, j, = r, F, j+ r, j Fj, By the Thrd Law ths sum becomes nt nt,, j+, j, =( r, r, j) F, j The vector r, r, j ponts rom the j th element to the th element. Central Forces: Internal Torques Cancel n Pars I the nternal orces between a par o partcles are drected along the lne jonng the two partcles then the torque due to the nternal orces cancel n pars. nt nt,, j+, j, =( r, r, j) F, j = 0 Ths s a stronger verson o Newton s Thrd Law than we have so ar used requrng that nternal orces are central orces. Wth ths assumpton, the total torque s just due to the external orces ext d = L However, so ar no solated tem has been encountered such that the angular momentum s not constant. F r r, j (,, j)
Angular Impulse and Change n Angular Momentum Angular mpulse Change n angular momentum Rotatonal dynamcs J = ( ) ave Δ t nt =ΔL t t ext t ext J = t ( ) ( ) ΔL L L = L L ( ) ( ) ext Concept Queston: Change n Angular Momentum A person spns a tenns ball on a strng n a horzontal crcle wth velocty v (so that the axs o rotaton s vertcal). At the pont ndcated below, the ball s gven a sharp blow (orce F ) n the orward drecton. Ths causes a change n angular momentum ΔL n the. ˆr drecton. ˆθ drecton 3. ˆk drecton Conservaton o Angular Momentum ext d Rotatonal dynamcs = L No external torques ext d 0 = = L Change n Angular momentum s zero ΔL L L = 0 ( ) ( ) Angular Momentum s conserved L = L ( ) ( ) o ar no solated tem has been encountered such that the angular momentum s not constant. 0 0 Concept Queston: Twrlng Person A woman,holdng dumbbells n her arms, spns on a rotatng stool. When she pulls the dumbbells nward, the moment o nerta changes and she spns aster. The magntude o the angular momentum about the vertcal axes passng through her center s. the same.. larger. 3. smaller. 4. not enough normaton s gven to decde. 3
Concept Queston: Fgure kater A gure skater stands on one spot on the ce (assumed rctonless) and spns around wth her arms extended. When she pulls n her arms, she reduces her rotatonal moment o nerta and her angular speed ncreases. Assume that her angular momentum s constant. Compared to her ntal rotatonal knetc energy, her rotatonal knetc energy ater she has pulled n her arms must be. the same.. Larger. 3. smaller. 4. not enough normaton s gven to decde. Table Problem : Rotatng Char and Wheel A person s sttng on a char that s ntally not rotatng and s holdng a spnnng wheel. The moment o nerta o the person and the char about a vertcal axs passng through the center o the stool s I,p, and the moment o nerta o the wheel about an axs, perpendcular to the plane o the wheel, passng through the center o mass o the wheel s I w = (/4)I,p. The mass o wheel s m w. uppose that the person holds the wheel as shown n the sketch such that the dstance o an axs passng through the center o mass o the wheel to the axs o rotaton o the stool s d and that md = (/3)I w. uppose the wheel s spnnng ntally at an angular speed ω s. The person then turns the spnnng wheel upsde down. You may gnore any rctonal torque n the bearngs o the stool. What s the angular speed o the person and stool ater the spnnng wheel s turned upsde down? Demo: Rotatng Wheel Bcycle Wheel and Rotatng tool () At rst the tran s started wthout the track movng. The tran and the track both move, one opposte the other. Demo: Tran () Then the track s held xed and the tran s started. When the track s let go t does not revolve untl the tran s stopped. Then the track moves n the drecton the tran was movng. (3) Next try holdng the tran n place untl the track comes up to normal speed (Its beng drven by the tran). When you let go the tran remans n ts statonary poston whle the track revolves. You shut the power o to the tran and the tran goes backwards. Put the power on and the tran remans statonary. A small gauge HO tran s placed on a crcular track that s ree to rotate. 4
Constants o the Moton When are the quanttes, angular momentum about a pont, energy, and momentum constant or a tem? No external torques about pont : angular momentum about s constant ext d 0 = = L No external work: mechancal energy constant 0 = W ext =ΔE mechancal No external orces: momentum constant ext dp F = Concept Queston: Conservaton Laws A tetherball o mass m s attached to a post o radus by a strng. Intally t s a dstance r 0 rom the center o the post and t s movng tangentally wth a speed v 0. The strng passes through a hole n the center o the post at the top. The strng s gradually shortened by drawng t through the hole. Ignore gravty. Untl the ball hts the post,. The energy and angular momentum about the center o the post are constant.. The energy o the ball s constant but the angular momentum about the center o the post changes. 3. Both the energy and the angular momentum about the center o the post, change. 4. The energy o the ball changes but the angular momentum about the center o the post s constant. Concept Queston: Conservaton laws A tetherball o mass m s attached to a post o radus R by a strng. Intally t s a dstance r 0 rom the center o the post and t s movng tangentally wth a speed v 0. The strng wraps around the outsde o the post. Ignore gravty. Untl the ball hts the post,. The energy and angular momentum about the center o the post are constant.. The energy o the ball s constant but the angular momentum about the center o the post changes. Experment 05: Moment o Inerta and Angular Collsons 3. Both the energy o the ball and the angular momentum about the center o the post, change. 4. The energy o the ball changes but the angular momentum about the center o the post s constant. 5
Experment 05: Goals Connect output o phototransstor to channel A o 750. Apparatus Measure the moment o nerta o a rgd body Investgate conservaton o angular momentum and knetc energy n rotatonal collsons. Measure and calculate non-conservatve work n an nelastc collson. Keep a copy o your results or the homework problem. Connect output o tachometer generator to channel B o 750. Connect power supply. Red button s pressed: Power s appled to motor. Red button s released: Rotor coasts: Read output voltage usng LabVIEW program. Use black stcker or tape on whte plastc rotor or generator calbraton. Calbrate Tachometergenerator pn motor up to ull speed, let t coast. Measure and plot voltages or 0.5 s perod. ample Rate: 5000 Hz. Count rotaton perods to measure ω. Program calculates average output voltage, angular velocty, and the calbraton actor angular velocty per volt perods. Rotor Moment o Inerta Plot only the generator voltage or rest o experment. Use a 55 gm weght to accelerate the rotor. 3 6
Analyss: Moment o Inerta Force and rotatonal equatons whle weght s descendng: mg T = ma Constrant: rt = I R α a = rα Rotatonal equaton whle slowng down = I R α peedng up lowng down olve or moment o nerta: rm(g rα ) + I R α = I R α I R = rm(g rα ) (α α ) Concept Queston: Understand Graph Output to Measure I R The angular requency along lne A-B s ncreasng because. the weght has ht the loor and s tenson n the strng s no longer applyng a torque.. the weght s descendng and the tenson n the strng s applyng a torque. 3. or some other reason. Concept Queston: Understand Graph Output to Measure I R The slope o the lne B-C s equal to. the angular acceleraton ater the weght has ht the loor. Measure I R : Results Reset sample rate: 500 Hz and 4 s, wtch rom Tachometer to Moment o Inerta. Measure and t best straght lne to get α, α, and I R : I R = rm(g rα ) (α α ). angular acceleraton beore the weght has ht the loor. 3. Nether o the above. 7
low Collson ample Rate tart Tme 000 Hz 4 sec low Collson Angular momentum: Angular Impulse: J z = rcton δt = I R α δt Fnd ω and ω, measure δt, t to nd α, α R, α. Calculate at t : at t + δt: L z, )ω L z, + I w )ω washer outer nner I = mr ( + r ) α w = ω δt Change n Angular momentum: I R α δt + I w )ω (I R )ω low Collson Fast Collson ample Rate 000 Hz tart Tme 4 sec Angle rotated through by rotor: Δθ rotor = ω δt + α Rδt = ω δt + ω ω δt δt = (ω + ω )δt Angle rotated through by washer: Δθ washer = α w δt = ω δt δt = ω δt Angle washer sld along rotor: Δθ rotor Δθ washer = ω δt Fnd ω (beore) and ω (ater), estmate δt or collson. Calculate washer = outer + nner I m( r r ) 8
Fast Collson Angular momentum: Knetc energy at dp: at t : at t + δt: L z, )ω L z, + I w )ω K = (I R + I w )ω R,mn Change n Angular momentum: Change n Knetc Energy: ΔL z + I w )ω (I R )ω ΔK = (I + I )ω R w (I )ω R 9