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

𝑚𝑔ℎ=1/2𝑚𝑣^2+12𝐼𝜔^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 𝑚𝑔ℎ with 𝑚m being the mass, 𝑔 the acceleration due to gravity, and ℎ 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 1/2𝐼𝜔^2, with 𝐼 being the moment of inertia (the object’s resistance to being rotated) and 𝜔 being the angular velocity.

For the hollow cylinder, the mass is placed the farthest from the center and thus 𝑟 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!

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I think they should reach the bottom at the same time