Chapter 11 Angular Momentum Analyss Model: Nonsolated System (Angular Momentum) Angular Momentum of a Rotatng Rgd Object Analyss Model: Isolated System (Angular Momentum)
Angular Momentum of a Partcle Consder a partcle of mass m located at the vector poston r and movng wth lnear velocty v Fgure (11.10) z L rp Fgure (11.10) x O r m p y The nstantaneous angular momentum L of the partcle relatve to the orgn O s defned as the cross product of the partcle s nstantaneous poston vector r and ts nstantaneous lnear momentum p : L rp The SI unt of angular momentum s kg m 2 /s. Both the magntude and the drecton of L depend on the choce of orgn. Rght-hand rule the drecton of L s perpendcular to the plane formed by r and p. Fgure (11.10) - r and p are n the xy plane, and so L ponts n the z drecton.
Because p = mv, the magntude of L s : L mvr sn where s the angle between r and p. L = 0 when r s parallel to p ( = 0 or 180 o ). In other words, when the lnear velocty of the partcle s along a lne that passes through the orgn, the partcle has zero angular momentum wth respect to the orgn. L = mvr when r s perpendcular to p ( = 90 o ). At that nstant, the partcle moves exactly as f t were on the rm of a wheel rotatng about the orgn n a plane defned by r and p.
Torque Is the product of the force F and poston vector r. The torque vector s n a drecton perpendcular to the plane formed by the poston vector and the force Example A partcle located at the poston vector r =(-1.0) + (2.0)j has a force F =(2.0) +(3.0)j actng on t. Calculate the torque vector about the orgn Lnear moton the net force on a partcle equals the tme rate of change of ts lnear momentum, F = dp / Rotatonal moton the net torque actng on a partcle equals the tme rate of change of ts angular momentum. We can show that the net torque act on a partcle equals the tme rate of change of ts angular momentum. dl d rp
dl τ Notes s vald only f : (1) and L are measured about the same orgn, (2) the expresson s vald for any orgn fxed n an nertal frame. Angular Momentum of a System of Partcles The total angular momentum of a system of partcles about some ponts s defned as the vector sum of the angular momenta of the ndvdual partcles : L L L 1 L2... n L where the vector sum s over all n partcles n the system. Concluson the total angular momentum of a system can vary wth tme only f a net external torque s actng on the system : τ ext dl d L dl That s : the tme rate of change of the total angular momentum of a system about some orgn n an nertal frame equals the net external torque actng on the system about that orgn.
Angular Momentum of a Rotatng Rgd Object Consder a rgd object rotatng about z axs. Each partcle rotates n xy plane about z axs wth an angular speed w. The angular momentum, L of a partcle of mass m L = p r = m v r = m r 2 w ; p = mv and v = r w The vector L s drected along the z axs as s the vector w. For the whole object, the angular momentum L = (m r 2 ) w = Iw
11.5) Conservaton of Angular Momentum Conservaton law n rotatonal moton : The total angular momentum of a system s constant n both magntude and drecton f the resultant external torque actng on the system s zero. From Equaton : τ ext dl Then : L = constant For a system of partcles the conservaton law s L n = constant, where the ndex n denotes the nth partcle n the system. If the mass of an object undergoes redstrbuton n some way the object s moment of nerta changes ts angular speed change because L = I. The conservaton of angular momentum s expressed n the form : L L constant (11.26) f If the system s an object rotatng about a fxed axs, such as the z axs : L z = I where L z s the component of L along the axs of rotaton and I s the moment of nerta about ths axs. 0
In ths case the conservaton of angular momentum s : I I constant (11.27) f f Ths expresson s vald both for rotaton about a fxed axs and for rotaton about an axs through the center of mass of a movng system as long as that axs remans parallel to tself. The net external torque s zero Theorem concernng the angular momentum of an object relatve to the object s center of mass : The resultant torque actng on an object about an axs through the center of mass equals the tme rate of change of angular momentum regardless of the moton of the center of mass. Ths theorem apples even f the center of mass s acceleratng, provded and L are evaluated relatve to the center of mass. In Equaton (11.26) thrd conservaton law to add. The energy, lnear momentum, and angular momentum of an solated system all reman constant : K U K f p = p f U f For an solated system L = L f
Example (11.9) : The Merry-Go-Round A horzontal platform n the shape of a crcular dsk rotates n a horzontal plane about a frctonless vertcal axle (Fgure (11.16)). The platform has a mass M = 100 kg and a radus R = 2.0 m. A student whose mass s m = 60 kg walks slowly from the rm of the dsk toward ts center. If the angular speed of the system s 2.0 rad/s when the student s at the rm, what s the angular speed when he has reached a pont r = 0.50 m from the center? M R m Fgure (11.16)
Example (11.11) : Dsk and Stck A 2.0-kg dsk travelng at 3.0 m/s strkes a 1.0-kg stck that s lyng flat on nearly frctonless ce (Fgure (11.18). Assume that the collson s elastc. Fnd the translatonal speed of the dsk, the translatonal speed of the stck, and the rotatonal speed of the stck after the collson. The moment of nerta of the stck about ts center of mass s 1.33 kg m 2. v d = 3.0 m/s Before After v df 2.0 m v s Fgure (11.18)