Ball a, of mass m a , is connected to ball b, of mass m b , by a massless rod of

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.

a) Find the ratio of the masses of the two balls. Express your answer numerically.

b) Find da, the distance from ball a to the system's center of mass. Express your answer in terms of L, the length of the rod.

Explanation / Answer

A:
if wetake axis as Axis a then Ia =mbL2
if we take axisas Axis b then Ib = maL2
thenIa/Ib = mbL2/maL2
giventhat Ia/Ib = 3
thenmbL2 /maL2  = 3
thenma/mb = 1/3
B:
Let ball A beat origin
thendistance of ma from origin is x1 = 0
then distanceof mb from origin is x2 = L
THENcentre of mass from the origin is
Xcm =(ma*x1 + mb*x2)/(ma +mb)
       = mb*L/ma +mb
butma/mb = 1/3
then mb= 3*ma
thenXcm = 3*ma*L/ma +3*ma = 3L/4

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