Equation For Rotational Inertia

Introduction:

Introduction:

If a body is in rest it cannot starts rotating itself along any axis and if it is in rotational motion it cannot stops itself. So there is an inertia in rotational motion. A quantity that measures the inertia of rotational motion is called rotational inertia or moment of inertia of the body. Rotational inertia plays the similar role in rotational motion as mass plays in linear motion. Rotational inertia is a scalar quantity.

Equation for Rotational Inertia:

The rotational inertia of a rotating body about a given axis is defined as the sum of products of masses of all the particles of the body and squares of their respective distances from the axis of rotation.

Mathematically, the rotational inertia I =Equation of Rotational inertia

Where, mi is the mass of any particle of elementary mass and ri is the distance of that elementary mass of the particle from axis of rotation. Rotational inertia depends upon the position of the axis of rotation, orientation of the axis of rotation, shape of the object, size of the object and distribution of mass of the object about the axis of rotation. Unit of rotational inertia is kg m2 or g cm2.

Equation of Rotational Inertia and Angular Momentum : Applications

(i) The angular velocity of a planet revolving in an elliptical orbit around the sun increases, when it comes nearer to the sun and the reverse is also true.When a planet revolving around the sun in an elliptical path near the sun, its rotational inertia about the axis through the sun decreases hence, angular speed increases. While on the other hand when the planet is far away from the sun the rotational inertia of the planet about the axis through the sun increases and hence angular speed decreases.

(ii) A girl who is perfect in ballet dancing can increase her angular velocity by folding her arms and bringing the stretched leg close to the other leg. When her hands and legs are stretched outwards her rotational inertia about the axis of rotation is large and hence the angular speed is quite small. By folding her arms and bringing the stretched leg close to the other leg, she decreases her rotational inertia so that the angular speed increases.

The moment of inertia, usually denoted I, measures the extent to which an object resists rotational acceleration about an axis, and is the rotational analogue to mass. Mass moments of inertia have units of dimension mass length2. It should not be confused with the second moment of area, which is used in bending calculations.

Geometrically simple objects have moments of inertia that can be expressed mathematically, but it may not be straightforward to symbolically express the moment of inertia of more complex bodies.

The following moments of inertia assume constant density throughout the object, and the axis of rotation is taken to be through the centre of mass, unless otherwise specified.

by: johnharmer

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