Question

You have two identical-looking metal spheres of the same size and the same mass. One is solid; the other is hollow. If you place them on a ramp, the gravitational torques that make them roll down the slope are the same. But one sphere has a greater angular acceleration, so it reaches the bottom of the ramp first. Which one gets to the bottom of the ramp first, and why?

Answer 1

The solid sphere will reach the bottom of the ramp first because it has a smaller moment of inertia, which allows for a greater portion of its gravitational potential energy to be converted into translational kinetic energy.

Step-by-step explanation:

When comparing a hollow sphere and a solid sphere rolling down an incline, the crucial difference is in how their mass is distributed. Both types of spheres have the same gravitational potential energy to begin with. As they roll down, this gravitational potential energy is transformed into both translational kinetic energy (KEtrans) and rotational kinetic energy (KErot). Because the solid sphere has a more compact mass distribution, it has a smaller moment of inertia than a hollow sphere of the same mass.

A smaller moment of inertia means that, for the same amount of gravitational energy, the solid sphere can allocate more energy into translational motion and less into rotational motion. Consequently, the solid sphere will have a greater angular acceleration and reach the bottom of the ramp first. This behavior holds true regardless of the objects' masses and sizes, as demonstrated by Galileo's experiments with inclined planes.