Question

One solid sphere \(\mathrm{A}\) and another hollow sphere \(\mathrm{B}\) are of the same mass and same outer radii. The moment of inertia about their diameters are respectively \(\mathrm{I}_{\mathrm{A}}\) and \(\mathrm{I}_{\mathrm{B}}\) such that... \(\\{\mathrm{A}\\} \mathrm{I}_{\mathrm{A}}=\mathrm{I}_{\mathrm{B}}\) \(\\{\mathrm{B}\\} \mathrm{I}_{\mathrm{A}}>\mathrm{I}_{\mathrm{B}}\) \(\\{\mathrm{C}\\} \mathrm{I}_{\mathrm{A}}<\mathrm{I}_{\mathrm{B}}\) $\\{\mathrm{D}\\}\left(\mathrm{I}_{\mathrm{A}} / \mathrm{I}_{\mathrm{B}}\right)=(\mathrm{d} \mathrm{A} / \mathrm{dB})$ (radio of their densities)

Short answer

The correct answer is Option C: \(\mathrm{I}_{\mathrm{A}}<\mathrm{I}_{\mathrm{B}}\).

Step-by-step solution

Determine the moment of inertia formula for a solid sphere

For a solid sphere (sphere A), the moment of inertia _(I)_ about any diameter can be found using the following formula: \[I_A = \frac{2}{5} m_A R_A^2\] where, - \(I_A\) = moment of inertia of sphere A - \(m_A\) = mass of sphere A - \(R_A\) = radius of sphere A

Determine the moment of inertia formula for a hollow sphere

For a hollow sphere (sphere B), the moment of inertia _(I)_ about any diameter can be found using the following formula: \[I_B = \frac{2}{3} m_B R_B^2 \] where, - \(I_B\) = moment of inertia of sphere B - \(m_B\) = mass of sphere B - \(R_B\) = radius of sphere B

Use the given information

According to the exercise, both spheres have the same mass and outer radii. So, \(m_A = m_B\) and \(R_A = R_B\). We can replace the mass and radii in the formulas we found in Steps 1 and 2 with the given information.

Compare the moment of inertia of both spheres

Now, we need to compare the moment of inertia of both spheres after replacing mass and radii in the formulas from Steps 1 and 2. For sphere A: \[I_A = \frac{2}{5} m_A R_A^2 = \frac{2}{5} m_B R_B^2\] For sphere B: \[I_B =\frac{2}{3} m_B R_B^2\] From the above two equations, it becomes clear that: \[\frac{2}{5} m_B R_B^2 < \frac{2}{3} m_B R_B^2\] Therefore, we can conclude that: \[ \mathrm{I}_{\mathrm{A}}<\mathrm{I}_{\mathrm{B}}\] Thus, the correct answer is

Option C

\(\mathrm{I}_{\mathrm{A}}<\mathrm{I}_{\mathrm{B}}\).