Rolling Down an Inclined Plane: Which Sphere Reaches the Bottom First?

Imagine you have two spheres of the same mass and size, but one is solid while the other is hollow. If both spheres are allowed to roll down an inclined plane, which one will reach the bottom first? This question delves into the fascinating world of rotational dynamics and how different mass distributions affect the motion of objects.

Rotational Dynamics and Roll Without Slipping

To understand the dynamics involved, let's first consider the condition of rolling without slipping. Both spheres need to maintain linear velocity vw middot;r (where w is the angular velocity, and r is the radius of the sphere).

Rotational Energy and Speed

Consider the rotational energy of each sphere. The solid sphere has a mass distributed close to its center, while the hollow sphere has a significant portion of its mass located far from the center. The rotational kinetic energy (KE) of an object is given by KErot1/2 middot; Iw2 (where I is the moment of inertia)

For the solid sphere, the moment of inertia Isolid2/5mr2. For the hollow sphere, the moment of inertia is Ihollow2/3mr2.

The total energy is given by mgh. In the case of the solid sphere, a larger part of this energy goes into translational motion, while in the hollow sphere, a larger part is used for rotational energy. Thus, the hollow sphere moves slower and the solid sphere reaches the bottom first.

Acceleration and Rotational Inertia

To determine which sphere will have more acceleration, we need to consider the ratio of the radius of gyration to the radius, which is essentially k/r. For the hollow sphere, this ratio is 2/5. For the solid sphere, the ratio would be different and generally smaller.

The hollow sphere has a higher angular inertia, meaning it takes more work to make it rotate. Because the mass is further away from the center, it is effectively on a longer lever, thus requiring more force to change its rotational state.

Friction's Role

It is important to note that if there were no friction at all, both spheres would reach the bottom at the same time. However, in real-world scenarios, friction is often present. Due to rotational inertia, the hollow sphere will have a slower acceleration compared to the solid sphere, which will reach the bottom first.