# Two spheres have identical radii and masses. How might you tell which of these spheres is hollow and which is solid?

Apr 26, 2016

For the same torque while rolling down an inclined plane, due to gravity, thin spherical shell will roll down slower due to lower angular velocity in comparison to the solid sphere.

#### Explanation:

You must have tried to find out if an egg was boiled or raw by spinning it on the counter gently. Stopping it suddenly by putting a finger on it and then releasing it immediately.

We use similar principle to solve your problem.

We know that the moment of inertia of a sphere about its central axis and a thin spherical shell (hollow sphere of your problem) is
Sphere, $I = \frac{2}{5} M {R}^{2}$.
Thin spherical shell, $I = \frac{2}{3} M {R}^{2}$, $M \mathmr{and} R$ being the mass and radius of the sphere respectively.
We see that spherical shell has more moment of inertia. (Check by making denominators equal for both.)

Now roll both the identical looking spheres down an inclined plane. Take care that neither of these slips.

The expression between torque $\tau$ and moment of inertia is given by
$\tau = I \frac{\mathrm{do} m e g a}{\mathrm{dt}}$, where $\omega$ is the angular velocity.

If same torque is provided to both while rolling down, due to gravity in this case, thin spherical shell will roll down slower due to lower $\omega$ in comparison to the solid sphere.