A small solid marble of mass m and radius r will roll without slipping along the loop-the-loop track if it is released from rest somewhere on the straight section of the track. For the following answers use m for the mass, r for the radius of the marble, R for the radius of the loop-the-loop and g for the acceleration due to gravity. A) From what minimum height h above the bottom of the track must the marble be released to ensure that is does not leave the track at the top of the loop.

Question

A small solid marble of mass m and radius r will roll without slipping along the loop-the-loop track if it is released from rest somewhere on the straight section of the track. For the following answers use m for the mass, r for the radius of the marble, R for the radius of the loop-the-loop and g for the acceleration due to gravity. A) From what minimum height h above the bottom of the track must the marble be released to ensure that is does not leave the track at the top of the loop.
B) If the marble is released from height 6R above the bottom of the track, what is the magnitude of the horizontal component of the force acting on it at point Q?

bobpursley · Answer

Take the energy from change of PE due to height (h-2r) to make total KE (translational and rotational). From that, you can solve for w. Change that expression to v/r, then set centripetal acceleation = mg, and you can then finally solve for the height it was dropped.