How High Must a Marble Track Be to Complete a Loop-the-Loop?

[dmahmoudi]

A marble rolls down a track and around a loop-the-loop of radius R. The marble has mass m and radius r. What minimum height h must the track have for the marble to make it around the loop-the-loop without falling off?

I'm stumped - I'm assuming I need to use energy for this problem, but how do I factor in the loop-the-loop. Maybe with rotational kinetic energy?

 

[George Jones]

A marble rolls down a track and around a loop-the-loop of radius R. The marble has mass m and radius r. What minimum height h must the track have for the marble to make it around the loop-the-loop without falling off?

I'm stumped - I'm assuming I need to use energy for this problem, but how do I factor in the loop-the-loop. Maybe with rotational kinetic energy?

There are two forces on the marble - the normal force applied by the track and the force of gravity. At the top of the track both forces point down towards the centre of the track. By Newton's second, the sum of these forces cause a centripetal acceleration. Use energy to find v and thus the centripetal acceleration at the top of the loop. Use Newton's second to find the normal force at the top. Find the height at which the normal force is zero. If released above this height, the marble goes loop-the-loop. If released below this height, the marble goes plop.

Regards,
George

 

[OlderDan]

A marble rolls down a track and around a loop-the-loop of radius R. The marble has mass m and radius r. What minimum height h must the track have for the marble to make it around the loop-the-loop without falling off?

I'm stumped - I'm assuming I need to use energy for this problem, but how do I factor in the loop-the-loop. Maybe with rotational kinetic energy?

You will need both rotational and translational kinetic energy to find the velocity of the marble as a function of its height relative to the starting height. You will then need to find the starting height that will give the marble the velocity at the top of the loop such that gravity will be just sufficient to provide the centripetal force needed for the circular path of the marble (i.e., there will be no normal force from the track at the top of the loop). Any lower velocity at the top would result in the marble being separated from the track by the gravitational force (separation would occur before the marble reached the top of the loop).

 

[dmahmoudi]

Awesome, so I see. Thanks for the help guys!