Question

Find the expression for the moment of inertia of a uniform rod of mass M, and length L, rotated about one of its ends.

Intergral you'll need to perform is given below

I = integral of ((r^2)(dm))

Answer 1

here,

dI = dm * x^2

Since the rod is uniform, the mass varies linearly with distance.

and dM = M/L * dx

Using the equation for dm, we substitute it into the first equation. Hence, we have:

dI = M/L * x^2 * dx

for a rod

the moment of inertia , I = integration(dI)

I = integration(x^2 * dx) from(-h) to ( L - h)

I = 1/3 * M * ( L^2 - 3 Lh + 3h^2)

When the rotation axis is at one end of the rod (h = 0), we have

I = 1/3 * M * L^2

so, the moment of inertia of rod about one end is I = 1/3 * M * L^2