Adams on a Roll

A group of hollow and solid cylinders once met at the top of a ramp. "Hey, solid cylinders," said the hollow cylinder, "Want to race?".

So, it began - a battle for the ages, and an outcome so impressive that we recreated it using an Adams simulation. 

As you can tell, the hollow cylinder did not do too well; it lost momentum early on in the descent and was left behind with the outcome never in doubt. Further confounding the issue was that the three solid cylinders were not identical. They had different sizes, material properties, weights, and that seemingly did not seem to impact the rate of their downward trajectory. They all finished alongside each other. We know for a fact (because Galileo told us) that objects dropped down from a certain height descend at the same rate regardless of their mass. (Assuming no aerodynamic losses, as my CFD brethren will point out). Does this apply to rolling objects too?

Well, at the risk of bringing back a painful high school memory, let me indulge you in a dynamicist's favorite vice- a free-body diagram.

Looking at the force components in the three dimensions, we have 

Where I is the moment of inertia and α is the angular velocity

By manipulating these equations, we can gather that 

We also know that the rotational inertias are different for a solid cylinder and hollow cylider

Assuming an infinitely thin wall we have for a hollow cylinder

and for a solid cylinder we have

Substituting this in the expression for acceleration derived above, we get 

This tells us that the cylinder's acceleration as it rolls down on an inclined ramp is independent of its mass, dimensions, or material density. It is only related to the inclined angle of the ramp, gravitational acceleration, and a constant imposed by its rotational moment of inertia. So as long as a cylinder is solid, heavier cylinders or lighter cylinders, shorter cylinders, or longer cylinders will navigate the ramp at the same rate. By extension, we have also learned that a hollow cylinder will always be slower than a solid cylinder, regardless of other factors.

So the next time if you think your neighborhood grocer shortchanged you on a can of beans you purchased, roll it down an inclined ramp and let physics do the rest.