MEAM 535 Advanced Dynamics

Fall 1999

Homework #2

Note that this homework is due on Thursday!
Submit only the first two problems on 9/23/99. The third one should be submitted on 9/28/99 along with HW #3.

1. Two cannon balls each of weight w are chained together and fired horizontally with a speed of v0 (read v sub zero) from the top of a wall of height h. The chain breaks during the flight, and one of the balls strikes the ground at time T after firing, at a distance d from the wall along, and distance a to the right of the line of fire. Determine the location of the other cannon ball at that instant.
   
   
2. A straight tube of uniform bore is rotating in a horizontal plane about a vertical axis through its center with uniform angular velocity omega. A small particle of mass m is free to slide in the tube. The coefficient of friction between it and the tube is mu.
   
   If at time t the particle is at radius r from the center of the tube, show that, neglecting the effect of gravity on the particle, its motion is given by the equation:
   
   If the length of the tube is 2a and the particle is projected into the end of the tube with a relative velocity v, under what conditions will the particle reach the center?
   
   
3. If A is a point of a rigid body A, B a point of rigid body b, A and B are in contact with each other at time t, and the velocities of A and B in any reference frame are equal to each other at time t, then bodies A and B are said to be rolling on each other at time t.
   
   When two rigid bodies A and B are rolling on each other, the angular velocity of B in A generally is not parallel to the plane P that is tangent to the surfaces of A and B at their points of contact with each other. When the angular velocity vector omega_B_in_A is parallel to P, the two bodies are said to be in pure rolling condition.
   
   The figure below shows a shaft terminating in a rounded truncated cone C enclosing an angle of 2*theta, this shaft being supported by a thrust bearing consisting of a fixed race B and four identical spheres of radius r. When the shaft rotates, rolling takes place at the points of contacts between B and S, as well as at the contact between S and C.
   
   To minimize wear, it is desired that S and C perform pure rolling motion on each other. Choose the dimension b as a function of r and theta to ensure this condition.