Question

(a) Find the moment of inertia, I, for a rod of length L and mass M for an arbitrary axis that is at distance of x from its one edge. (b) Now find the moment of inertia when the axis is at one edge. (c) Find the moment of inertia when the axis is in the middle.

Please leave detailed steps

Answer 1

consider a small length dx from a rod of length L

let O be the mid point of it such on either sides it is having a length of L/2

moment of inertia due to small length be dI

then

dI = rho * dx * x^2

total moment of inertia I = integration of dI = rho * integration of x^2 dx

I = Rho * (X^3)/3 from - L/2 to +L/2

I = rho/3 * (L/2)^3 -(-L/2)^3)

I = rho *L^3/12

as rho = linear mass density = mass/L

I = ML^2/12

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part B

dI - rho *x^2 dx

I = o to L   rho dx x^2

I = rho * )x^3/3) o to l

I = rho * l l^2/3

I = Ml^2/3

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