The figure shows a ball with mass m = 0.405 kg attached to the end of a thin rod with length L = 0.516 m and negligible mass. The other end of the rod is pivoted so that the ball can move in a vertical circle. The rod is held horizontally as shown and then given enough of a downward push to cause the ball to swing down and around and just reach the vertically up position, with zero speed there.

(a) What initial speed must be given the ball so that it reaches the vertically upward position with zero speed?
(b) What then is its speed at the lowest point?
(c) What then is its speed at the point on the right level with the initial point?
(d) If the ball's mass were doubled, would the answers to (a) through (c) increase, decrease, or remain the same?

Can I start with h = L - L*cos(theta) = L(1 - cos(theta))? Where do I go from there?

3 answers

(a) Use conservation of energy. To reach the top with zero velocity
(1/2) M Vo^2 = M*g*L
Vo = sqrt(2 g L)

(b) At the lowest point,
(1/2)MV^2 = (1/2) M Vo^2 + M g L
V^2 = Vo^2 + 2 g L
= 4 g L
V = 2 sqrt(g L)

(c) What do you think? The P.E. is the same.

(d) Note that M does not appear in any of the answers above. What does that tell you?

Got it, thanks very much!

wot m8