Question

a solid cylinder with radius r has the same mass as a solid sphere of radius r. if the cylinder and sphere have the same moment of inertia about their centers, what is the sphere's radius?

Answer 1

The sphere's radius is 4.47cm

How to determine the radius of the sphere?

Let r represent the radius of the cylinder, and R symbolize the radius of the sphere.

Given:

Radius of the cylinder
`(\(r\))` = 4.0 cm

We know the moment of inertia
`(\(I\))` for a solid cylinder about its center is given by:

`\[ I_{\text{cylinder}} = (1)/(2) m r^2 \]`

And the moment of inertia for a solid sphere about its center is given by:

`\[ I_{\text{sphere}} = (2)/(5) m R^2 \]`

Since both the cylinder and sphere have the same mass, we can equate the expressions for their moments of inertia:

`\[ (1)/(2) m r^2 = (2)/(5) m R^2 \]`

Given
`\(r = 4.0\) cm`, let's solve for
`\(R\)` (the radius of the sphere):

`\[ (1)/(2) * (4.0 \, \text{cm})^2 = (2)/(5) R^2 \]`

`\[ 8.0 \, \text{cm}^2 = (2)/(5) R^2 \]`

Now, solve for
`\(R^2\)`:

`\[ R^2 = \frac{8.0 \, \text{cm}^2 * 5}{2} \]`

`\[ R^2 = 20.0 \, \text{cm}^2 \]`

Finally, take the square root to find
`\(R\)`:

`\[ R = \sqrt{20.0 \, \text{cm}^2} \]`

`\[ R \approx 4.47 \, \text{cm} \]`

Complete question:

A solid cylinder with a radius of 4.0 cm has the same mass as a solid sphere of radius R. If the cylinder and sphere have the same moment of inertia about their centers, what is the sphere's radius?