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220 Practice Exam 3 Spring 2015 Lea & Burke Ch 8, 10, 14
Potential Energy and Non-conservative forces, Oscillations, and Collisions.
1. A mass m on an ideal vertical spring with constant k is compressed a distance x below its
equilibrium position and released from rest. The mass then comes to rest a height h above equilibrium.
(a) Draw a free-body-diagram for the mass when it and the compressed spring are released from rest.
(b) What is maximum speed of the mass vMAX from the time of release to when it reaches height h?
Express vMAX in terms of m, k, x, and any constants.
(c) Express the height h in terms of m, k, x, and any constants.
2. A ball of mass m on the end of a string of length L is pulled to one side and released from rest at a
height h above the lowest position allowed by the string. When the ball is released, it follows a circular
path back to its lowest position, where its speed is vf .
(a) Draw a free-body-diagrams for the mass at the top and bottom of its swing.
(b) Express vf in terms of h and any constants.
(c) What is the tension T in the string when the ball reaches its lowest position?
Express T in terms of h, m, L, and any constants.
3. A mass m on a frictionless surface is attached to two springs on either side with spring constants k1
and k2. When the mass is at x = 0, the springs are at their equilibrium positions. The mass is displaced
to x = A and released from rest.
(a) Draw a free-body-diagram for the mass at x = A. Assume spring 1 is stretched and spring 2
compressed at this position.
(b) Use newton's second law and the relationship between acceleration and displacement for simple
harmonic motion to derive the period of oscillation of the mass.
(c) Graph the potential energy U versus displacement x of the mass between x = -A and x = A.
4. A mass m with initial velocity v0 slides to rest up a plane inclined at angle θ.
The mass slides a distance D < v02/(2gsinθ) before coming to rest.
(a) Draw a free-body-diagram of the mass as it slides up the plane.
(b) Graph the mass's kinetic energy K versus distance x as it slides up the plane from x = 0 to x = D.
(c) What is the coefficient of friction μ between the mass and plane?
5. Cart 1 of mass m is initially moving with speed v east when it collides completely inelastically with
cart 2 of mass 3m initially moving with speed v/2 to the west.
(a) What is the kinetic energy lost in the collision?
(b) Draw before and after diagrams for the two carts using velocity vectors of corrects lengths.
(c) Redraw the diagrams in the center of mass frame.
(d) What is the final kinetic energy in the center of mass frame?
6. Cart 1 of mass m is initially moving with speed v east when it collides perfectly elastically with cart
2 of mass 3m initially moving with speed v/2 to the west.
(a) What are the final velocities of the two carts?
(b) Draw before and after diagrams for the two carts using velocity vectors of corrects lengths.
(c) Redraw the diagrams in the center of mass frame.
(d) How does the total kinetic energy in the center of mass frame compare to total kinetic energy of the
“lab” frame of parts (a) and (b)?
7. Two stars of equal mass M and radius R are in initially at rest separated by a distance D > 2R.
(a)With what speed will they eventually collide if no other forces act on them?
(b) Graph the total potential energy U of the two stars versus separation distance x as they accelerate
towards each other from initial separation x = D to final separation x = 2R when they collide.
(c)With what speed v would they need to orbit their common center of mass to remain in uniform
circular motion separated by D?