Problem 3/62 Page 146, Engineering Mechanics - Dynamics, Meriam and Kraige, 4th Edition
Given: A ball of mass m at the end of a string of length L. The ball is being twirled in a vertical plane.
Find: The minimum speed the ball must have to maintain a positive tension in the string at the uppermost position of the ball. The tension in the string when the ball is at its lowest position assuming the ball is traveling at the aforementioned minimum speed.
Parameter Values:
m = 50 g = 0.05 kg
L = 1 m
1. Mechanical System: Ball at uppermost position
2. Free Body Diagram: Not included. Would show ball, weight force vertically downward, tension force vertically downward. The tension force would just be zero at the minimum speed. The coordinate axes would be shown adjacent to the free body diagram. The n direction would be normal to the motion, directed toward the center of the circle (vertically downward). The t direction would be tangent to the circle.
3. Equations:
S F = ( m g + T ) n = m g
n
m a = m ( at t + an n ) = m ( dv/dt t +
v2/r n ) = m ( dv/dt t
+ v2/L n )
4. Solve:
Considering the n direction:
m g = m v2/L
Solving for v:
v = ( L g )1/2 = 3.13 m/s
1. Mechanical System: Ball at lowermost position
2. Free Body Diagram: Not included. Would show ball, weight force vertically downward, tension force vertically upward. The coordinate axes would be shown adjacent to the free body diagram. The n direction would be normal to the motion, directed toward the center of the circle (vertically upward). The t direction would be tangent to the circle.
3. Equations:
S F = ( -m g + T ) n = (-m
g + T ) n
m a = m ( at t + an n ) = m ( dv/dt t +
v2/r n ) = m ( dv/dt t
+ v2/L n ) = m ( dv/dt t + g n )
4. Solve:
Considering the n direction:
-m g + T = m g
Solving for T:
T = 2 m g = 0.981 N