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Angular Momentum & Torque for Systems of Particles Lecturer: Professor Stephen T. Thornton Reading Quiz A particle is located in the xy-plane at a location x = 1 and y = 1 and is moving parallel to the +y axis. A force is exerted on the particle along the +x axis. L and t are in what directions about the origin? A) B) C) D) E) O L and t are along the +z axis. L and t are along the -z axis. L is along the +z axis; t is along the –z axis. L is along the -z axis; t is along the +z axis. L is along the +y axis; t is along the +x axis. · y v F x z Answer: C L = r´ p y r x p · · F t Last Time Angular momentum Vector (cross) products Torque again with vectors Today Angular momentum and torque system of particles rigid objects Unbalanced torque Kepler’s 2nd law Atwood Machine. An Atwood machine consists of two masses, mA 7.0 kg and mB 8.2 kg connected by a cord that passes over a pulley free to rotate about a fixed axis. The pulley is a solid cylinder of radius R0 0.40 m and mass 0.80 kg. (a) Determine the acceleration a of each mass. (b) What percentage of error in a would be made if the moment of inertia of the pulley were ignored? Ignore friction in the pulley bearings. System of Particles The angular momentum of a system of particles can change only if there is an external torque—torques due to internal forces cancel. This equation is valid in any inertial reference frame. It is also valid about a point uniformly moving in an inertial frame of reference. We are starting to get very technical! System of Particles The equation above is not valid in general about a point accelerating in an inertial frame of reference. But the center of mass is special! The equation is true even for an accelerating center of mass of a system of particles or for a rigid object: Angular Momentum for a Rigid Object For a rigid object, we can show that its angular momentum when rotating around a particular axis is given by: Add up all the particles. If L is along a symmetry axis (z here) through CM, particles on one side of symmetry axis cancel L on the other side. rotating Li L Iω dL dt axis I So we finally have these equations for a rigid object. The values must be calculated about 1) Origin or axis fixed in an inertial frame. or 2) An origin at the CM or about an axis passing through the CM. If we do not have this, then things get real complicated! We have reached our limit here!! Torque and Angular Momentum Vectors dL dt L τt r ω Torque Gravity and Extended Objects Gravitational torque acts at the center of mass, as if all mass were concentrated there: Torque Gravity and Extended Objects Gravitational torque acts at the center of mass, as if all mass were concentrated there. Do the Falling Rigid Body demo again. Conceptual Quiz You are looking at a bicycle wheel along its axis. The wheel rotates CCW and is supported by a string attached to the rear of the handle. When the wheel is released, the end of the handle closest to you will A) B) C) D) move up move to the left move to the right move down Do bicycle wheel demo. Answer: C Move to the right. The picture below is looking from above. Li L Lf L t Torque r Lf Li L Lf Top view Li Conceptual Quiz A man sits at rest on a frictionless rotating stool. He holds a rotating bicycle wheel that has an angular momentum L directed up. When he flips the wheel over, so that it has L directed down, the angular momentum of the system (man + stool + wheel) is A) B) C) D) E) zero. L, up. L, down. 2L, up. 2L, down. Answer: B Angular momentum has to be conserved. There is no torque to change it. Do experiment. Angular Momentum and Torque for a Rigid Object A system that is rotationally imbalanced will not have its angular momentum and angular velocity vectors in the same direction. A torque is required to keep an unbalanced system rotating. L and in p in An unbalanced car wheel will cause problems on your wheel bearings. We need to keep our wheels well balanced, dynamically not just statically. Kepler’s 2nd Law 1 dA r (v dt sin ) 2 dA 1 rv sin but L r mv mrv sin dt 2 dA L constant dt 2m There is no torque so L is constant, and Kepler’s second law states that each planet moves so that a line from the Sun to the planet sweeps out equal areas in equal times. Conceptual Quiz You are holding a spinning bicycle wheel while standing on a stationary turntable. If you suddenly flip the wheel over so that it is spinning in the opposite direction, the turntable will: A) remain stationary B) start to spin in the same direction as before flipping C) to spin in the same direction as after flipping Conceptual Quiz You are holding a spinning bicycle wheel while standing on a stationary turntable. If you suddenly flip the wheel over so that it is spinning in the opposite direction, the turntable will: The total angular momentum of the system is L upward, and it is conserved. So if the wheel has −L downward, you and the table must have +2L upward. A) remain stationary B) start to spin in the same direction as before flipping C) start to spin in the same direction as after flipping Conceptual Quiz Two different spinning disks have the same angular momentum, but disk 1 has more kinetic energy than disk 2. A) disk 1 B) disk 2 C) not enough info Which one has the bigger moment of inertia? See hint on next slide. Disk 1 Disk 2 Conceptual Quiz Two different spinning disks have the same angular momentum, but disk 1 has more kinetic energy than disk 2. A) disk 1 B) disk 2 C) not enough info Which one has the bigger moment of inertia? 1 KE = 2 I 2 = L2 / (2 I) (used L = I ). Disk 1 Disk 2 Conceptual Quiz Two different spinning disks have the same angular momentum, but disk 1 has more kinetic energy than disk 2. A) disk 1 B) disk 2 C) not enough info Which one has the bigger moment of inertia? 1 KE = 2 I 2 = L2 / (2 I) (used L = I ). Because L is the same, bigger I means smaller KE. Disk 1 Disk 2