Classical Mechanics Review 3, Units 1-16 Today: Review for midterm #3

v1,i - v2,i = - (v1, f - v2, f ) FNet = MaCM τNet = Iα Important Equations 1. Elastic Collisions: 2. System of particles – Rigid bodies: Dynamics: Relation between Angular and Linear quantities: Energy: Parallel Axis Theorem: I = ICM + MD2 Rolling motion: v1,i - v2,i = - (v1, f - v2, f ) FNet = MaCM τNet = Iα vCM = R aCM = αR

Example: Elastic Collision Two masses approach each other with equal and opposite velocities as measured in the lab reference frame. The mass moving to the right is twice as massive as the one moving to the left. The collision between them is elastic.   What is the velocity of the center of mass before the collision? What are the velocities of the two objects after the collision? What is the velocity of the center of mass after the collision? g v1,i - v2,i = - (v1, f - v2, f )

Example: Rotating Rod, I A rod of length L and mass M is attached to a frictionless pivot and is free to rotate in the vertical plane. The rod is released from rest in the horizontal position. The moment of inertia of the rod about the center of mass is I = ML2/12 (a) What is the moment of inertia of the rod about the pivot? (b) What is the initial angular acceleration? (c) What is the initial tangential acceleration of its center of mass? I = ICM + MD2 τNet = Iα = MgL/2

Example: Rotating Rod, II A rod of length L and mass M is free to rotate on a pin. The rod is released from rest in the horizontal position. The moment of inertia of the rod about the center of mass is I = ML2/12 (a) What is its angular speed when the rod reaches its lowest position? (b) Determine the tangential speed of the center of mass and of the lowest point on the rod when it is in the vertical position.

Example: Rolling up a Plane A uniform sphere of mass M and radius R rolls without slipping up a plane inclined at an angle  with the horizontal. The initial center-of-mass velocity of the sphere is v. ICM = 2MR2/5. (a) What is the frictional force exerted by the plane on the sphere? (b) What is the maximum height, H, above the horizontal, the sphere can reach?  R M V fs 

Example: Atwood's Machine with Massive Pulley A pair of masses are hung over a massive disk-shaped pulley as shown. Find the acceleration of the blocks. y x m2g m1g a T1 T2  R M m1 m2 For the hanging masses use F = ma -m1g + T1 = -m1a -m2g + T2 = m2a For the pulley use  = I T1R - T2R (Since for a disk)

Example: Billiard Balls A white billiard ball with mass m = 1.65 kg is moving directly to the right with a speed of v = 3.22 m/s and collides with a black billiard ball with the same mass that is initially at rest. The two collide elastically and the white ball ends up moving at an angle above the horizontal of θw = 41° and the black ball ends up moving at an angle below the horizontal of θb = 49°. Find the final speeds of the balls and the final total energy of the system.

Example: Falling Rod (a) The rotational kinetic energy of the rod. A rod of length L and mass M is pivoted about a horizontal, frictionless pin through one end. The rod is released, almost from rest in a vertical position. The moment of inertia of the rod about its center is I = ML2/12. At the instant the rod makes an angle of θ with the vertical find: (a) The rotational kinetic energy of the rod. (b) The angular acceleration of the rod. (c) The speed of the center of mass of the rod. L m M θ P

Example: Pulley and Mass A wheel of radius R, mass M, and moment of inertia I is mounted on a frictionless horizontal axle. A light cord wrapped around the wheel supports an object of mass m. Find the angular acceleration of the wheel, the linear acceleration of the object and the tension T. FNet = mg – T = ma τNet = Iα = TR a = αR

Example: Pulley and horizontal mass A block of mass m1 = 1 kg sits atop a plane with coefficient of kinetic friction 0.2 and is connected to mass m2 = 2 kg through a string that goes over a pulley of mass M = 4 kg and radius R = 0.2 m. The pulley rotates about its axis without friction and the string moves over the pulley without slipping. The system starts at rest and mass m2 falls through a height H = 2 m. The moment of inertia of the pulley is I = MR2/2   What is the velocity of mass m2 just before it hits the ground? H = 2 m H m2 = 2  kg M, R Rpul = m1 m2