A solid marble, a solid disk and a hoop have the same mass (M = 0.9 kg) and the

Question

A solid marble, a solid disk and a hoop have the same mass (M = 0.9 kg) and the same radius. Each object is held at rest at the same height and then released. At the bottom of the ramp is a loop-the-loop of radius 1 m. What is the minimum height that the marble, disk and hoop can be released so that all three will safely make it around the loop-the loop? At the top of the loop-the-loop, what is the force the track exerts on the hoop when released at the minimum height?

Explanation / Answer

First let us write the energy balance between total mechanical energy at the initial position and total mechanical energy at the top of the loop: Initially at height H the energy is purely potential energy: m g H During the motion, the marble gains translational and rotational kinetic energy: Etrans = 1/2 m v^2 Erot = 1/2 I w^2 = 1/2 (2/5 m r^2) (v/r)^2 = 1/5 m v^2 So the energy balance, when starting from rest at height H and going to height 2R (at the top of the loop) is: m g H = m g (2 R) + 1/2 m v^2 + 1/5 m v^2 m g H = 2 m g R + 7/10 m v^2 which equates the initial potential energy on the left (kinetic energy zero initially) to the total mechanical energy on the right (potential energy + translational kinetic energy + rotational kinetic energy). To stay on the loop at the highest point, the velocity must be such that the gravity alone is not enough to provide the centripetal force (there has to be a nonzero normal force helping too): m v^2 / R > m g m v^2 >m g R Using this on the right hand side of the energy balance we have m g H > 2 m g R + 7/10 ( m g R) m g H > 27/10 m g R H > 27/10 R So the minimum height is 2.7 times the radius of the loop.

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