Entrance Examination in USA part 1

(Wisconsin University)

1. A marble bounces down stairs in a regular manner, hitting each step at the same place and bouncing the same height above each step (see figure). The stair height equals its depth (tread = rise) and the coefficient of restitution e is given.  Find the necessary horizontal velocity and bounce height. (The coefficient of restitution is defined as e = -v_f/v_i, where v_f and v_i are the vertical velocities just after and before the bounce. respectively).

2.A projectile of mass m is shot (at velocity v) at a target of mass M, with a hole containing a spring of constant k. The target is initially at rest and can slide without friction on a horizontal surface. Find the distance  ∆x that the spring compresses at maximum.

(CUSPEA)

3.A pendulum of mass m and length l is released from rest in a horizontal position. A nail a distance d below the pivot causes the mass to move along the path indicated by the dotted line. Find the minimum distance d in terms of l such that the mass will swing completely round in the circle shown in the figure.( Wisconsin University)

answer: d = 3l/5

4.A thin uniform stick of mass m with its bottom end resting on a frictionless table is released from rest an angle  θ_o to the vertical. Find the force exerted by the table upon the stick at an infinitesimally small time after its release.(University California at Berkeley)

5.Two steel spheres, the lower of radius 2a and the upper of radius a are dropped from a height h (measured from the center of the larger sphere) above a steel plate as shown. Assume that the centers of the spheres always lie on a vertical line and all collisions are elastic, what is the maximum height of the upper sphere will reach.
Hint: Assume that the larger sphere collides with the plate and recoils before it collides with the smaller sphere.(Wisconsin University)

6. A mass M slides without friction on the roller coaster track shown in figure. The curved sections of the track have radius of curvature R. The mass begins its descent from the height h. At some value of h, the mass will begin to lose contact with the track. Indicate on the diagram where the mass loses contact with the track and calculate the minimum value of h for which this happens! (Wisconsin University)

Answer :  h_min = 3R/4

7. A bowling ball of uniform density is thrown along a horizontal alley with initial velocity V₀ in such a way that it initially slides without rolling. The ball has mass m, coefficient of static friction μ_s and coefficient of sliding friction μ_d with the floor. Ignore the effect of air friction. Compute the velocity of the ball when it begins roll without sliding. (Princeton University)

Answer: v = 5/7 V₀

8. Calculate the minimum coefficient of friction necessary to keep a thin circular ring  from sliding as it rolls down a plane inclined at an angle θ with respect to the horizontal plane. (Wisconsin University)

Answer: μ = ½ tanθ

9. A billiard ball of radius  R and mass M is struck with a horizontal cue stick at a height h above the billiard table as shown in figure. Given that the moment of inertia of a sphere is 2/5 MR² , find the value of h for which the ball will roll without slipping. (Wisconsin University)

10. Assume all surfaces to be frictionless and the inertia of pulley and cord negligible (see figure). Find the horizontal force necessary to prevent any relative motion of m_1, m₂ and M.
(Wisconsin University)