Solid and hollow cylinders both roll from rest down an inclined plane. Explain the difference in speeds at the bottom.

Wouldn't it depend on their masses also? And radius?

To determine the difference in speeds at the bottom of an inclined plane between solid and hollow cylinders, we need to consider their rotational and translational motion.

First, let's understand the concept of rotational and translational motion. When an object rolls down an inclined plane, it has a combination of rotational and translational motion. Translational motion refers to the linear movement of the center of mass of the object, while rotational motion refers to the object spinning about its axis.

Now, let's look at the differences between solid and hollow cylinders:

1. Mass distribution: In a solid cylinder, the mass is evenly distributed throughout its volume. However, in a hollow cylinder, there is a significant portion of the mass missing from the center. The hollow space creates a difference in the mass distribution.

2. Moment of inertia: Moment of inertia accounts for an object's resistance to rotational motion. The moment of inertia for a solid cylinder is higher than that of a hollow cylinder with the same mass and dimensions. This is because the mass is distributed farther away from the axis of rotation in a solid cylinder, creating more resistance to rotational motion compared to a hollow cylinder.

Considering these differences, the solid cylinder will have a higher rotational inertia and, therefore, a slower rotation compared to the hollow cylinder. However, the translational motion of both cylinders will be the same, assuming there are no other factors like friction or air resistance.

As a result, at the bottom of the inclined plane, the solid cylinder will have a lower speed due to its higher rotational inertia. The hollow cylinder, with a lower rotational inertia, will have a higher speed compared to the solid cylinder.

To summarize, the difference in speeds at the bottom of an inclined plane between a solid and hollow cylinder arises due to the difference in their rotational inertias, where the solid cylinder has a higher rotational inertia, resulting in a slower speed, while the hollow cylinder has a lower rotational inertia, leading to a higher speed.