Prove Rotational Inertia of Uniform Sphere is 2/5MR^2

[bluejay18]

How would I prove that the Rotational Inertia of a unform sphere is 2/5M(R)squared, about any axis?I have no idea where to start. . .

 

[vaishakh]

how did you findratational inertia of others?
reply to this first


i think you migh be knowing the method to find sphere volume. similarly consider sphere as a overlapping of inginite cylinders.

 

[charng]

To prove that the rotational inertia of a uniform sphere is 2/5MR^2, we can use the formula for rotational inertia, I = ∫r^2dm, where r is the distance from the axis of rotation and dm is the mass element.

For a uniform sphere, the mass element dm can be represented as dm = ρdV, where ρ is the density of the sphere and dV is the volume element.

Next, we can express the volume element as dV = πr^2dh, where h is the height of the sphere.

Substituting these expressions into the formula for rotational inertia, we get I = ∫r^2ρπr^2dh.

We can then simplify this to I = πρ∫r^4dh.

Since the sphere is uniform, ρ is constant and can be taken out of the integral. This leaves us with I = πρ∫r^4dh = πρr^4∫dh.

Integrating with respect to h, we get ∫dh = h.

Substituting the height of the sphere, h = 2R, we get I = πρr^4(2R) = 2πρR^5.

Finally, we can substitute the density of the sphere, ρ = M/V, where M is the mass and V is the volume, to get I = 2πMR^5/V.

Since the volume of a sphere is V = (4/3)πR^3, we can further simplify this to I = (2/5)MR^2.

Therefore, we have proven that the rotational inertia of a uniform sphere is 2/5MR^2 about any axis.