Question

Ball a, of mass ma

, is connected to ball b, of mass mb
, by a massless rod of length L
. (Figure 1)The two vertical dashed lines in the figure, one through each ball, represent two different axes of rotation, axes a and b. These axes are parallel to each other and perpendicular to the rod. The moment of inertia of the two-mass system about axis a is Ia
, and the moment of inertia of the system about axis b is Ib
. It is observed that the ratio of Ia
to Ib
is equal to 3:
Ia/Ib=3
Assume that both balls are pointlike; that is, neither has any moment of inertia about its own center of mass.
1. Find the ratio of the masses of the two balls.
2. Find da, the distance from ball a to the system's center of mass

Answer 1

To find the ratio of the masses of the two balls connected by a rod, use the parallel-axis theorem. To find the distance from ball a to the system's center of mass, use the formula for the center of mass of a system of objects.

Step-by-step explanation:

To find the ratio of the masses of the two balls, we can use the parallel-axis theorem. The moment of inertia of the two-mass system about axis a is given as Ia, and the moment of inertia about axis b is given as Ib. We are given that the ratio of Ia to Ib is 3, so we can write the equation:

Ia/Ib = ma*da^2 / mb*db^2 = 3

To find the distance from ball a to the system's center of mass, da, we can use the formula for the center of mass of a system of objects:

da = (mb*L)/(ma+mb)