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Moment of inertia equation and formulas of rigid objects

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What is moment of inertia?

Moment of inertia is defined as:”The sum of the products of the mass of each particle of the body and square of its perpendicular distance from axis.”

Or :

The product mass and the square of perpendicular distance from the axis of rotation is known as moment of inertia.

Moment of inertia formula

 

I = mr²

For a rigid body moving about a fixed axis,the laws of motion have the same form as those of rectilinear motion,with moment of inertia replacing mass,angular replacing linear velocity,angular momentum replacing linear momentum,etc.Hence the kinetic energy of a body rotating about a fixed axis with angular velocity ω is ½ω²,which corresponds to ½mv² for the kinetic energy of a body of mass m translated with velocity v.See also Routh’s rule;theorem of parallel axes.

Moment of inertia equation

Consider a mass m attached to the end of a massless rod.Let us assume that the bearing at the pivot point O is frictionless.Let the system be in a horizontal plane.A force F is acting on the mass perpendicular to the rod and hence this will accelerate the mass according to:

F = ma

In doing so the force will cause the mass to rotate about O.Since tangential acceleration is related to angular

acceleration α by the equation.

angular acceleration = rα

As turning effect is produced by torque τ ,it would therefore ,be better to write the equation for rotation in terms of torque.This can be done by multiplying both sides of the above equation by r.Thus

rF = τ = torque = mr²α

Which is rotational analogue of the Newton’s second law of motion.

Here F is replaced by τ,a by α and m by mr².The quantity mr² is known as the moment of inertia and is represented by I.

Importance of moment of inertia

The moment of inertia plays the same role in angular motion as the mass in linear motion.It may be noted that moment of inertia depends not only on mass m but also on r².

list of moments of inertia

  • Moment of inertia of Hoop

formula of moment of inertia of cylindrical shell

  • Moment of inertia of a disc

moment of inertia of a disc

  • Moment of inertia of a solid sphere

moment of inertia of solid sphere

  • Moment of inertia of a hollow cylinder

moment of inertia of hollow cylinder

  • Moment of inertia of thin rod

moment of inertia of thin rod

  • Moment of inertia of a rectangle

moment of inertia of rectangle

 

  • Moment of inertia of a long thin rod

moment of inertia of thin rod with rotation axis through ends

  • Moment of inertia of a spherical shell

moment of inertia of spherical shell

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