Rolling Objects on a Slope: Sphere vs Hoop
Which object will rise farther up the slope, a solid sphere or a thin hoop?
If these two objects roll up the slope without slipping, which will rise farther up the slope?
A. The sphere
B. The hoop
C. Both the same
D. More information about the objects' mass and diameter is needed
Answer
If these two objects roll up the slope without slipping, the one that will rise farther up the slope is the hoop.
This situation is based on the fact that the solid sphere and the hoop have the same mass. The analysis can be done by observing the behaviors of the objects while rolling up the slope. The key factor here is the moment of inertia of each object.
It should be noted that if the energy of rotation is not taken into account, one might expect that both objects would rise to the same height. However, by applying the conservation of energy and taking into consideration the total initial kinetic energy and the final potential energy, the situation changes.
When the radius and mass of the objects cancel out in the conservation of energy equation, the thin hoop will have a larger moment of inertia compared to the solid sphere. This difference in moment of inertia allows the hoop to rise farther up the slope compared to the sphere.