ME 232 Homework Problems

Problem 3/143 Page 182, Engineering Mechanics - Dynamics, Meriam and Kraige, 4th Edition

Given: Collar of mass m sliding on a frictionless track.  The track is composed of a quarter circle of radius R and a straight horizontal segment of length L.  The collar is released from rest on the vertical portion of the quarter circle and travels to the end of the horizontal segment.  A spring of unstretched length L0 and stiffness k is attached to the collar and to the center of the circular arc.

Find: The speed of the collar when it reaches the end of the straight segment.

Parameter Values:
m = 3 kg
R = 0.8 m
L = 0.6 m
L0 = 0.4 m
k = 200 N/m

1. Mechanical System: Collar as it travels from release point to end point.

2. Free Body Diagram: Not included.  Would show collar, weight of collar acting vertically downward, spring force directed from the collar to the other spring attachment point, and a normal force from the track.

3. Equations:
Work = - m g Dh (gravity) - 1/2 k (sf2 - si2) (spring) + 0 (force normal to motion)
          =  m g R - 1/2 k { [ (R2+L2)1/2 - L0 ]- (R-L0)2 }

D Kinetic Energy = 1/2 m (vB2 - 0)

4. Solve:

 m g R - 1/2 k { (R-L0)2 - [ (R2+L2)1/2 - L0 ]2 } = 1/2 m vB2
vB = { 2 g R - k/m  { [ (R2+L2)1/2 - L0 ]- (R-L0)2 }}1/2 = 1.537 m/s