Question

Ball a, of mass m_a, is connected to ball b, of mass m_b, by a massless rod of length L. The two vertical dashed lines in the figure, one througheach 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-masssystem about axis a is I_a, and the moment of inertia of the system about axis b is I_b. It is observed that the ratio of I_a to I_b is equal to 3:

$\frac{I_a}{I_b} = 3$

Assume that both balls are pointlike; that is, neither has anymoment of inertia about its own center of mass.

Q. Find the ratio of the masses of the two balls.

(Express your answernumerically.)

$\frac{m_{\rm a}}{m_{\rm b}}$ = ?

Q. Find d_a, the distance from ball A to the system's center of mass.

(Express your answer in terms of L,the length of the rod.)

Answer 1

PARTA:

if wetake axis as Axis a then I_a =m_bL²

if we take axisas Axis b then Ib = m_aL²

thenIa/Ib = m_bL²/m_aL²

giventhat Ia/Ib = 3

thenm_bL² /m_aL²  = 3

thenm_a/m_b = 1/3

PARTB:

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

butm_a/m_b = 1/3

then mb= 3*ma

thenXcm = 3*ma*L/ma +3*ma = 3L/4

Answer 2

Wouldn't the mass of A be 3 times greater?

IF Ia = 3Ib

And I = MR²

then it would be 3MbR²