A small object enters the top A of a circular path with horizontal speed vQ. Treating the object as a particle and neglecting friction, determine an expression containing Vo, g, and R for the angle ? at which the object separates from the path and becomes a projectile. After doing so, find the numerical value of the angle ? for Uo = 0. Solution when moving along the path -N + m g cos theta = m v^2/r leaves when N = 0 so m g cos theta = mv^2/r m v^2 = m g R cos theta conservation of energy 1/2 m v0^2 + m g R = 1/2 m v^2 + m g R cos theta 1/2 mv0^2 + m g R = 1/2 m g R cos theta + m g r Cos theta 1/2 m v0^2 + m g R = 3/2 m g R cos theta v0^2 + 2 g R = 3 g R cos theta cos theta = (v0^2 + 2 g R)/( 3 g R) b) when v0 = 0 cos theta = 2/3 theta = arcos(2/3)= 48.19 degrees .

1. A small object enters the top A of a circular path with horizontal speed vQ. Treating the object as
a particle and neglecting friction, determine an expression containing Vo, g, and R for the angle
? at which the object separates from the path and becomes a projectile. After doing so, find the
numerical value of the angle ? for Uo = 0.
Solution
when moving along the path
-N + m g cos theta = m v^2/r
leaves when N = 0
so m g cos theta = mv^2/r
m v^2 = m g R cos theta
conservation of energy
1/2 m v0^2 + m g R = 1/2 m v^2 + m g R cos theta
1/2 mv0^2 + m g R = 1/2 m g R cos theta + m g r Cos theta
1/2 m v0^2 + m g R = 3/2 m g R cos theta
v0^2 + 2 g R = 3 g R cos theta
cos theta = (v0^2 + 2 g R)/( 3 g R)
b) when v0 = 0
cos theta = 2/3
theta = arcos(2/3)= 48.19 degrees