Two cylinders of the same mass and shape, one hollow and one solid, are set on incline and allowed to roll down. Which cylinder will reach the bottom of the incline first? Why?

1 Answer
Jul 21, 2015

Solid will reach the bottom first.


Let's analyze this from the point of view of the law of conservation of energy.

In the beginning, the potential energy for both cylinders at the top of an incline is the same since their masses are the same.

Then they are rolling down with some linear speed of their centers of gravity and angular speed of rotation. At the end of the movement the potential energy, again, is the same.

That means that the kinetic energy acquired during their motion should be the same as well.

Kinetic energy of a cylinder has two parts - one related to the forward movement of its center of gravity and another - related to its rotation. For a rolling down cylinder the angular speed of rotation and linear speed of its center of gravity are linearly related: if an angular speed is A (radians/sec) and radius of a cylinder is R (meters) than linear speed would be the distance in meters it covers in 1 second, that is the length of an arc RA. So, linear speed is V=RA (meters/sec).

If the angular speed of rotation for both cylinder is the same, their linear speed of forward movement is the same as well, and the kinetic energy of the forward movement is the same.

Kinetic energy of rotation, however, is different for these cylinders. With the same angular speed the hollow cylinder has more kinetic energy of rotation because its mass distribution is close to its surface and, therefore, more particles are moving with greater speed on a greater radius. So, to equalize this effect, the angular speed of rotation of the hollow cylinder must be smaller.

So, the hollow cylinder should be slower, the solid - faster.