Question

What is the moment of inertia of a pyramid or a cone ?

Answer 1

The mass of the elemental disc is dm=ρ⋅πr2dz

The density of the cone is

ρ=MV=M13πR2h

Therefore,

dm=M13πR2hπr2dz

dm=3MR2hr2dz

But

Rr=hz

r=Rzh

dm=3MR2h⋅R2h2⋅z2dz=3Mh3z2dz

The moment of inertia of the elemental disc about the z−axis is

dI=12dmr2

dI=12⋅3Mh3z2⋅z2R2h2dz

dI=32⋅MR2h5z4dz

Integrating both sides,

I=32⋅MR2h5∫h0z4dz

I=32⋅MR2h5[z55]h0

I=32⋅MR2h5⋅h55

=310MR2

Answer 2

The moment of inertia is a physical quantity which describes how easily a body can be rotated about a given axis. It is a rotational analogue of mass, which describes an object's resistance to translational motion. Inertia is the property of matter which resists change in its state of motion. Inertia is a measure of the force that keeps a stationary object stationary, or a moving object moving at its current speed. The larger the inertia, the greater the force that is required to bring some change in its velocity in a given amount of time. Suppose a heavy truck and a light car are both at rest, then intuitively we know that more force will be required to push the truck to a certain speed in a given amount of time than will be needed to push the car to that same speed in the same amount of time.