Question

Three point masses M1 M2 and M3 are located at the vertices of an equilateral triangle of length a . the moment of inertia of system about an Axis along the altitude of the triangle passing through M1 is ??

Answer 1

see the attachment,
NN' is the altitude passing through m1 .
we know,
moment of inertia is the product of mass and square of separation between particle and axis of rotation .
e.g , M.I = mr²

here, we see, separation of mass m1 and altitude NN' is 0 .
alteration between mass m2 and NN' is (a/2) also for m3 separation is (a/2)

now,
moment of inertia about altitude passing through m1 = I1 + I2 + I3
where I1 , I2 , and I3 are M.I of m1 , m2 and m3 respectively .

hence,
M.I = m1.(0) + m2(a/2)² + m3(a/2)²
= a²/4 [m2 + m3 ]

Answer 2

refer to the attachment mate