Circular motion question

Question

A ball of mass M attached to a string of length L moves in a vertical plane. At the top of the circular path, the tension in the string is twice the weight of the ball. At the bottom, the ball just clears the ground. Air resistance is negligible. Express all answers in terms of M, L and g.

(a) What is the magnitude and direction of the net force on the ball when it is at the top?

(b) What is the speed, v, of the ball at the top?

The string is cut when the ball is at the top of its circular path.

(c) What is the time it takes the ball to reach the ground?

(d) What is the horizontal distance the ball travels before hitting the ground?

Answer 1

a) If you draw the FBD of the problem, you easily see that the net force, F_net = Mg + 2(w); where w = mg= weight ==> F_net = Mg + 2Mg = 3Mg, downward direction!

b) This a circular motion problem, this means that F_net is also a centripetal force

==> F_net = mv²/L ==> 3Mg = Mv²/L, solving for the speed, v yields

v = √(3Lg)

c) using the kinematic equation x = v₀t +.5at², but v₀ = 0 and a = g, x = 2L

==> t = √(2x/g) or t = 2√(L/g)

d) to find the horizontal distance, use the formula: x = vt