Animated Solution for Physics - Rotational Motion: Consider two uniform discs of same thickness and different radii $R_1=R$ and $R_2=\alpha R$ made of the same material. If the ratio of their moments of inertia $I_1$ and $I_2$ respectively, about their axes is $I_1:I_2=1:16$, then the value of $\alpha$ is

Imagine you are holding two uniform discs. They are cut from the exact same sheet of metal, which means they share the same material density $\rho$ and the same thickness $t$. However, they are not identical in size. The first disc has a radius $R_1=R$, while the second disc is larger, with a radius $R_2=\alpha R$.

Our goal is to find this unknown multiplier $\alpha$, given that the ratio of their moments of inertia about their central axes is $I_1:I_2=1:16$.

To solve this, we need to express the moment of inertia in terms of the given parameters. The standard formula for the moment of inertia of a uniform disc about its central axis is:

But we don't know the mass $M$ directly. We do know that mass is the product of volume and density. For a disc, the volume is its face area multiplied by its thickness.

Now, let's substitute this mass back into our inertia formula. This is a crucial step because it reveals how inertia scales with the radius when thickness and density are constant.

Notice the beautiful result: the moment of inertia is proportional to the fourth power of the radius!

Now, let's write this expanded inertia expression for both of our discs.

For the first disc:

For the second disc:

We are given the ratio of their inertias. Let's divide $I_1$ by $I_2$. Because both discs share the same $\pi$, $\rho$, and $t$, all these constants elegantly cancel out, leaving us with a pure relationship between the radii.

The problem states that the ratio $\frac{I_1}{I_2}$ is exactly $\frac{1}{16}$. Let's equate our derived ratio to this given value.

This simplifies to:

To find $\alpha$, we just need to take the fourth root of 16. Since $2\times 2\times 2\times 2=16$, we have: