Define moment of inertia.

Moment of Inertia:

The moment of inertia (denoted by I ) is a measure of an object's resistance to changes in its rotational motion about an axis. It depends on the mass distribution of the object relative to the axis of rotation. Mathematically, it is expressed as:

I = ∑ mi ri^2

where:

- mi is the mass of a small element of the object,

- ri is the perpendicular distance of that element from the axis of rotation.

- ^2 power of 2

The moment of inertia can be different for the same object depending on the axis around which it is rotating.

Example:

Consider a thin rod of length L  and mass M , rotating about an axis perpendicular to its length and passing through one of its ends.

The moment of inertia for this rod is given by:

I = 1/3ML^2

Question:

A solid cylinder of mass 10 kg and radius 0.5 m is rotating about its central axis. Calculate the moment of inertia of the cylinder.

Solution:

For a solid cylinder rotating about its central axis, the moment of inertia is given by:

I = 1/2MR^2

where:

- M = 10kg  (mass of the cylinder),

- R = 0.5 m (radius of the cylinder).

Substitute the given values:

I = 1/2 * 10 * (0.5)^2

I = 1/2 * 10 * 0.25

I = 1/2 * 2.5 = 1.25 kg .m^2

Answer: The moment of inertia of the cylinder is  1.25 kg.m^2 .