If I have a solid rod and hollow rod with the same mass and I let them roll down a ramp, which one reaches the bottom first and why?

Solid, Hollow Rod Down Ramp

When an object is rolling down a ramp, its energy is made up of three components:

π‘šπ‘”β„Ž=1/2π‘šπ‘£^2+12πΌπœ”^2

The first term is the potential energy; this is the energy is takes to lift the object up the ramp. This is equal to π‘šπ‘”β„Ž with π‘šm being the mass, 𝑔 the acceleration due to gravity, and β„Ž the height of the ramp.

The second term is the translational kinetic energy; this is the energy it takes for the object to move down the ramp.

The third term is the rotational kinetic energy; this is the energy it takes for the object to roll. This is equal to 1/2πΌπœ”^2, with 𝐼 being the moment of inertia (the object’s resistance to being rotated) and πœ” being the angular velocity.

For the hollow cylinder, the mass is placed the farthest from the center and thus π‘Ÿ is large, 𝐼 is large, and consequently it is slow. For the case of the solid cylinder, the mass distribution occurs more closer to the center, thus has lower 𝐼 and highest velocity.

Thus, the solid cylinder reaches the bottom first.

If you want to practice more questions like this, check out these other moment of inertia problems!

Here are some other tricky questions:

I think they should reach the bottom at the same time