A point mass `m` starts from rest and slides down the surface of a frictionless hemisphere of radius r as shown figure. Measure angle from the vertical and potential energy from the top. Find
a. Find the changes in potential energy of the mass with angle
b. Find the kinetic energy as a function of angle
c. Find the radial and tangential acceleration as a function of angle
d. Find the angle at which the mass files off the hemisphere
e. If there is friction between the mass and hemisphere, does the mass fly off at a greater or lesser angle than in part (d) ?

Answer

(a) `-mgR(1-cos theta)` (b) `mgR(1-cos theta)` (c) `a_("radial")=v^(2)//R, a_(t)=g sin theta`