Question

6.A dumbbell consists of two uniform spheres of mass M and radius R joined by a thin rod of mass m. The distance between the centers of the spheres is l as shown in the figure. What is the moment of inertia of this device about an axis through the center of the rod perpendicular to the rod? About an axis along the rod?

Answer 1

1. The moment of inertia of this device about an axis through the center of the rod perpendicular to the rod (axis A).

The moment of inertia of a composite body is equal to the sum of the moments of its individual pieces,

$$I_A=2I_{sphere}+I_{bar}$$

For axis A, the rod is rotating about its centre of mass. Each sphere is a

distance $R+\frac{L}{2}$ from the axis of rotation, so we must use the parallel axis theorem.

$$I_A=2\left(M\left(R+\frac{L}{2}\right)^2+\frac{2}{5}MR^2\right)+\frac{1}{12}mL^2$$

2. About an axis along the rod (Axis C).

The bar and both spheres are rotating about their own centres when rotating about axis C.

$$I_A=2\left(\frac{2}{5}MR^2\right)+0=\frac{4}{5}MR^2$$