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Classical Mechanics Review 3, Units 1-15

Classical Mechanics Review 3, Units 1-15. Today: Review for midterm #3. Important Equations. 1. Elastic Collisions: 2. Perfectly Inelastic Collisions: 3. System of particles – Rigid bodies: Dynamics: Relation between Angular and Linear quantities :

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Classical Mechanics Review 3, Units 1-15

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  1. Classical MechanicsReview 3, Units 1-15 Today: Review for midterm #3

  2. Important Equations • 1. Elastic Collisions: • 2. Perfectly Inelastic Collisions: • 3. System of particles – Rigid bodies: • Dynamics: • Relation between Angular and Linear quantities: • Parallel Axis Theorem:I = ICM + MD2 v1,i-v2,i= - (v1, f-v2, f ) FNet= MaCM τNet= Iα

  3. Example: Elastic Collision g • 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 completely 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? v1,i-v2,i = - (v1, f-v2, f )

  4. 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.

  5. Example: Rotating Rod • 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 = MR2/2 • (a)What is the moment of inertia of the rod about its left end? •   (b) What is the initial angular acceleration? Repeat with the rod making an angle of θbelow the horizontal. • (c) What is the initial tangential acceleration of its center of mass? I = ICM + MD2 τNet= Iα= MgL/2 τNet= Iα= MgLcosθ/2

  6. Example: Collision • A ball of mass mand a ball of mass M each hang from very light strings of length L. At rest the balls hang side by side, barely touching. The m ball is pulled to the left until the angle between its string and the vertical is 90.0o. The Mball stays at rest at the bottom. The mball is released from rest and collides with the Mball. Right after the collision the speed of the Mball is V. • (a) What is the speed of the mball at the bottom of its swing? • (b) What is the velocity of the mball right after the collision ? • (c) Is the collision elastic? v1,i-v2,i=? - (v1, f-v2, f )

  7. 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 For the pulley use =I m2g (Since for a disk) m1g M • For the hanging masses use F = ma • -m1g + T1=-m1a • -m2g+T2=m2a  R T2 T1 a m2 m1 T1R - T2R a

  8. Example: Collision on a Vertical Spring • A vertical spring with k is standing on the ground supporting a m2block. A m1block is placed at a height hdirectly above the m2block and released from rest. After the collision the two blocks stick together. • (a) What is the speed of the two blocks right after the collision? • (b) What is the maximum compression of the spring? • (c) What is the position of the equilibrium after the collision?

  9. Example: System of Two Objects Two objects, one having twice the mass of the other, are initially at rest. A force F pushes the mass Mto the right. A force 2F pushes the mass 2M upward. What is the magnitude of the acceleration of the center of mass? 2M M 2F F

  10. Example: Two Ice-Skaters y x V • Two ice-skaters of mass m1and m2, each having an initial velocity ofviin the directions shown, collide and fall and slide across the ice together. The ice surface is horizontal & frictionless. • 1. What is the speed of the skaters after the collision? • 2. What is the angle θ relative to the x axis that the two skaters travel after the collision?  y x

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