Why Do Spherical Balls of Different Sizes Roll at the Same Speed Downhill?

[dhphysics]

Homework Statement

Suppose two spherical balls roll down a hill without slipping. If both are released from rest, which one will roll the fastest?

The answer is that both will roll at the same speed, even if they are of different sizes and weights, but I do not understand why

Homework Equations

Conservation of mechanical energy E= mgh + (1/2)m*v^2 + (1/2)I*ω^2

The Attempt at a Solution

I would think that the smallest ball rolls the fastest. If all of the gravitational potential energy mgh is turned into kinetic energy (1/2)m*v^2 + (1/2)I*ω^2, then the bigger ball would lose more energy to rotational kinetic energy because the rotational inertia I is greater because the radius is bigger. However, that is not the correct answer. I think that friction is not considered in this problem.

 

[Dick]

Homework Statement

Suppose two spherical balls roll down a hill without slipping. If both are released from rest, which one will roll the fastest?

The answer is that both will roll at the same speed, even if they are of different sizes and weights, but I do not understand why

Homework Equations

Conservation of mechanical energy E= mgh + (1/2)m*v^2 + (1/2)I*ω^2

The Attempt at a Solution

I would think that the smallest ball rolls the fastest. If all of the gravitational potential energy mgh is turned into kinetic energy (1/2)m*v^2 + (1/2)I*ω^2, then the bigger ball would lose more energy to rotational kinetic energy because the rotational inertia I is greater because the radius is bigger. However, that is not the correct answer. I think that friction is not considered in this problem.

You aren't thinking very hard about this. Assume they are all uniformly dense spheres. Now find a relation between I and the mass m and R and a relation between ω and v and R. Now what do you think?