 # 375 physics/spheres.p

## Description

This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

# 375 physics/spheres.p

Two spheres are the same size and weight, but one is hollow. They are
each made of uniform material, though of course not the same material.
With a minimum of apparatus, how can I tell which is hollow?

physics/spheres.s

Since the balls have equal diameter and equal mass, their volume and
density are also equal. However, the mass distribution is not equal,
so they will have different moments of inertia - the hollow sphere has
its mass concentrated at the outer edge, so its moment of inertia will
be greater than the solid sphere. Applying a known torque and observing
which sphere has the largest angular acceleration will determine which
is which. An easy way to do this is to "race" the spheres down an
inclined plane with enough friction to prevent the spheres from sliding.
Then, by conservation of energy:

mgh = 1/2 mv^2 + 1/2 Iw^2

Since the spheres are rolling without sliding, there is a relationship
between velocity and angular velocity:

w = v / r

so

mgh = 1/2 mv^2 + 1/2 I (v^2 / r^2) = 1/2 (m + I/r^2) v^2

and

v^2 = 2mgh / (m + I / r^2)

From this we can see that the sphere with larger moment of inertia (I) will
have a smaller velocity when rolled from the same height, if mass and radius
are equal with the other sphere. Thus the solid sphere will roll faster.

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