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