Question

A small solid sphere and a small thin hoop are rolling along a horizontal surface with the same translational speed when they encounter a 20° rising slope. 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.

Short answer

The correct option is (b).

Step-by-step solution

Rotational kinetic energy 

The total kinetic energy is equal to the sum of rotational kinetic energy and translational kinetic energy.Here, both are moving at the same linear speed; you have to find out which one has a greater rational kinetic energy.

Both the solid sphere and the thin hoop have the same translation speed.

The angle of the inclined plane is \(\theta = {20^ \circ }\).

You can assume both to have the same mass and radius.

Explanation

As they have the same translational speed, both have the same translational kinetic energy. The magnitude of the angular velocity is the same for both objects as they have the same radius.

You also know that the thin hoop has a greater moment of inertia than the solid sphere of the same mass and radius.

Therefore, the thin hoop has larger kinetic energy, i.e., the total kinetic energy of the thin hoop is greater.

When the solid sphere and thin hoop reach the height point, the total kinetic energy converts into gravitational potential energy. Therefore, the final potential energy is equal to the initial total kinetic energy. As the gravitational potential energy is proportional to the object's height from the ground, the thin hoop rises further up the slope.