Explain the concept of moment of inertia.

The moment of inertia is a measure of an object's resistance to rotational motion.

When an object rotates around an axis, it has a tendency to resist changes in its rotational motion. This resistance is known as the moment of inertia. It is a property of the object that depends on its mass distribution and the axis of rotation. The moment of inertia is denoted by the symbol I and is measured in units of kg m².

The moment of inertia of a point mass is given by the formula I = mr², where m is the mass of the object and r is the distance from the axis of rotation. For a system of point masses, the moment of inertia is the sum of the individual moments of inertia.

For continuous objects, such as a rod or a disk, the moment of inertia is calculated using calculus. The moment of inertia of a rod of length L and mass M rotating about an axis through its centre of mass is given by the formula I = (1/12)ML². The moment of inertia of a disk of radius R and mass M rotating about an axis through its centre of mass is given by the formula I = (1/2)MR².

The moment of inertia plays an important role in rotational dynamics. It is used to calculate the angular acceleration of an object when a torque is applied to it. The relationship between torque, moment of inertia, and angular acceleration is given by the formula τ = Iα, where τ is the torque, α is the angular acceleration, and I is the moment of inertia.