Question

Calculate the moment of inertia for two spheres of radius \(10\,\text{cm}\) placed \(20\,\text{cm}\) apart.

• A. \(\frac{7}{5}MR^2\)
• B. \(\frac{14}{5}MR^2\)
• C. \(\frac{9}{5}MR^2\)
• D. \(2MR^2\)

Hint:

For systems of multiple bodies, calculate the moment of inertia of each body using the parallel axis theorem and then add them to obtain the total moment of inertia.

Answer 1

The Correct Option is B

Concept: The moment of inertia of a body measures its resistance to rotational motion about a given axis. For composite systems, the total moment of inertia is obtained by adding the moments of inertia of individual parts. For a solid sphere, the moment of inertia about its own central axis is: \[ I_{\text{sphere}} = \frac{2}{5}MR^2 \] When the axis of rotation does not pass through the center of mass of the body, we use the parallel axis theorem: \[ I = I_{\text{cm}} + Md^2 \] where \(I_{\text{cm}}\) = moment of inertia about the center of mass \(d\) = distance between the axis and the center of mass.

Step 1: Moment of inertia of each sphere about its center. \[ I_{\text{cm}} = \frac{2}{5}MR^2 \]

Step 2: Distance between the spheres. The distance between the centers is \(20\,\text{cm}\). If the axis is taken through the midpoint between the spheres, each sphere is at a distance \(R\) from the axis. Thus, \[ d = R \]

Step 3: Applying the parallel axis theorem. \[ I = I_{\text{cm}} + Md^2 \] \[ I = \frac{2}{5}MR^2 + MR^2 \] \[ I = \frac{7}{5}MR^2 \] This is the moment of inertia for one sphere about the given axis.

Step 4: Total moment of inertia of two spheres. \[ I_{\text{total}} = 2 \times \frac{7}{5}MR^2 \] \[ I_{\text{total}} = \frac{14}{5}MR^2 \] Thus, the moment of inertia of the system is: \[ \frac{14}{5}MR^2 \]