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 ]2
- (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 ]2 - (R-L0)2 }}1/2 = 1.537
m/s