Two uniform solid balls, one of radius R and mass M, the other of radius 2R and mass 8M, roll down a high incline. They start together from rest at the top of the incline. Which one will reach the bottom of the incline first?
Question
Two uniform solid balls, one of radius R and mass M, the other of radius 2R and mass 8M, roll down a high incline. They start together from rest at the top of the incline. Which one will reach the bottom of the incline first?
Solution
The speed of a rolling object down an incline is independent of its mass, so the mass of the balls does not affect the time it takes for them to reach the bottom of the incline. However, the speed does depend on the radius of the object.
The moment of inertia for a solid sphere is (2/5)MR^2, where M is the mass and R is the radius. The larger sphere has a moment of inertia of (2/5)(8M)(2R)^2 = (32/5)MR^2, while the smaller sphere has a moment of inertia of (2/5)MR^2.
The acceleration of a rolling object down an incline is given by a = gsin(theta)/(1+I/MR^2), where g is the acceleration due to gravity, theta is the angle of the incline, I is the moment of inertia, M is the mass, and R is the radius.
For the larger sphere, a = gsin(theta)/(1+(32/5)), and for the smaller sphere, a = gsin(theta)/(1+(2/5)).
Therefore, the smaller sphere will reach the bottom of the incline first because it has a larger acceleration.
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