Question

Find moment of inertia about axis shown which is equidistant from both spheres

Answer 1

Step 1: Understanding the problem setup.
The two spheres are placed along a line, and the axis of rotation is equidistant from both spheres. We need to calculate the total moment of inertia of the system about this axis. 

Step 2: Moment of inertia of a single sphere.
The moment of inertia \(I\) of a sphere about an axis through its center is given by: \[ I_{\text{sphere}} = \frac{2}{5}mR^2 \] where \(m\) is the mass of the sphere and \(R\) is the radius of the sphere. 

Step 3: Using the Parallel Axis Theorem.
The axis is not through the center of either sphere, so we must apply the Parallel Axis Theorem to shift the axis to the point where the moment of inertia is needed. The Parallel Axis Theorem states that the moment of inertia about any axis parallel to the center of mass axis is: \[ I = I_{\text{cm}} + md^2 \] where \(d\) is the distance from the center of mass of the sphere to the new axis. Since the axis is equidistant from both spheres, let the distance between the center of each sphere and the axis be \(d\). The moment of inertia for each sphere about the given axis is: \[ I_{\text{sphere}} = \frac{2}{5}mR^2 + md^2 \] 

Step 4: Total Moment of Inertia.
Since there are two spheres, the total moment of inertia about the axis is the sum of the moments of inertia of both spheres: \[ I_{\text{total}} = 2 \left(\frac{2}{5}mR^2 + md^2\right) \] 

Final Answer:
The total moment of inertia of the system about the given axis is: \[ I_{\text{total}} = 2 \left(\frac{2}{5}mR^2 + md^2\right) \]