A ball starts from rest on top of a smooth spherical surface of radius R, as shown. The tangential velocity of the ball is given by the expression
v t = { 2 g R ( 1 - cos q ) } 0.5  where g is the acceleration due to gravity and q is angle that the ball makes at the center of the sphere with respect to the y - axis . Find the angle , q, at which the ball flies off the spherical surface.

a ) 48.12 o
b ) 41.81 o
c ) 33.69 o
d ) 49.23 o

 

Let N be the normal reaction on the ball; M the mass of the ball ; v t, the tangential velocity ; a t, the tangential acceleration.

Since there is no net normal acceleration , the equation of motion in the normal direction is given by

The ball flies off the spherical surface when it makes an angle of 48.12 o at the center of the sphere with respect to the y axis.