The figure below shows a ball with mass m = 0.334 kg attached to the end of a thin rod with length L = 0.522 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? m/s
(b) What then is its speed at the lowest point? m/s
(c) What then is its speed at the point on the right level with the initial point? m/s
If the ball's mass were doubled, would the answers to (a) through (c) increase, decrease, or remain the same?
https://i.stack.imgur.com/Uoefv.gif
Is it:
a) mGl = .334 * 9.8 * .522?
b) ans to a, but negative?
c) not sure.
d) stay the same, since everything proprtional?
a) is wrong. $mgL$ is the potential energy difference, not a speed. You write conservation of energy $$\frac{1}{2}mv_i^2+mgh_i=\frac{1}{2}mv_f^2+mgh_f$$ and you have $$h_f-h_i=L$$ Note that $v_f=0$, so $$v_i^2=2gL$$
b) Once again use the conservation of energy as in my first equation, but now $$h_f-h_i=-L$$ Since you now know $v_i$, you can calculate $v_f$
c) conservation of energy and $h_f=h_i$ means that $v_f^2=v_i^2$, but since on the left is moving down, and on the right is moving up, then $v_f=-v_i$
d) you are right. Mass does not affect the result, since both kinetic and potential energy are proportional to the mass