Consider a rod , two particles of mass m1 , m2 are at its two ends then

distance of center of mass from m1 is

r1= m2r/(m1+m2)                      (using formula)

distance of center of mass from m2 is
r2 = m1r/(m1+m2)          .............(2)

now moment of inertia of two particles is

I1 = m1r1²& I2 = m2r2²
I = I1+I2 = m1r1² + m2r2²   ........(3)

from 2 & 3 above expression becomes

I = m1m2r²/(m1+m2)

To determine the moment of inertia of the system, we need to consider the individual moments of inertia of the masses m1 and m2 about an axis perpendicular to the rod and passing through the center of mass.



The moment of inertia of m1 can be calculated using the formula: I1 = m1 * r1^2, where r1 is the distance of m1 from the axis of rotation.

In this case, the distance r1 is half of the length of the rod, as the center of mass of the rod is at its center. Thus, r1 = l/2.

Therefore, the moment of inertia of m1 is: I1 = m1 * (l/2)^2 = m1 * l^2/4.



Similar to m1, the moment of inertia of m2 can be calculated using the formula: I2 = m2 * r2^2, where r2 is the distance of m2 from the axis of rotation.

In this case, the distance r2 is also half of the length of the rod, as the center of mass of the rod is at its center. Hence, r2 = l/2.

Therefore, the moment of inertia of m2 is: I2 = m2 * (l/2)^2 = m2 * l^2/4.



The total moment of inertia of the system is the sum of the individual moments of inertia of m1 and m2. Therefore, the equation becomes:

I_total = I1 + I2 = m1 * l^2/4 + m2 * l^2/4

Simplifying this equation, we get:

I_total = (m1 + m2) * l^2/4

Thus, the moment of inertia of the system about an axis perpendicular to the rod and passing through the center of mass is given by (m1 + m2) * l^2/4.


In summary, the moment of inertia of a system with two masses attached to a light rod can be determined by calculating the individual moments of inertia of the masses and summing them up. The moment of inertia of each mass is given by m * r^2, where m is the mass and r is the distance from the axis of rotation. The total moment of inertia of the system is obtained by adding the individual moments of inertia. In this case, the moment of inertia is (m1 + m2) * l^2/4, where m1 and m2 are the masses of the two particles and l is the length of the rod.