Consider a rigid object rotating about a fixed axis at a certain angular velocity. Since every particle in the object is moving, every particle has kinetic energy. To find the total kinetic energy related to the rotation of the body, the sum of the kinetic energy of every particle due to the rotational motion is taken.
Let us assume there are n particles, K.E. is represented as

Let us consider a body like a sphere, disc or a wheel rolling on a plane surface without slipping. A rolling body undergoes translational motion about the centre of mass and rotational motion about an axis passing through the centre of mass.
Total K.E. of a rolling body = Transitional K.E. + Rotational K.E.
Where m = mass of the body
R = Radius of the body
v = Translational speed of the body
I = M.I. of the body about an axis of rotation passing through the centre of mass.
K = Radius of gyration of the body

Radius of gyration of a given body about a given axis of rotation is the normal distance of a point from the axis, where if whole mass of the body is placed then its moment of inertia will be exactly same as it has with its actual distribution of mass. or
S.I. Unit of radius of gyration is metre.
The value of radius of gyration shall depend upon shape and size of the body, position and configuration of the axis of rotation and also on distribution of mass of the body w.r.t. the axis of rotation.

By comparing linear motion and rotational motion formulae we find that role played by moment of inertia in rotational motion is same as the role played by mass in translational motion.
In translational motion, mass of a body represents its inertia, ie reluctance to undergo a change in its state of rest or of uniform translational motion. Obviously, moment of inertial also represents inertia of a body in rotational motion (i.e. its reluctance to undergo a change in its state of rotation).Moment of inertia is also called rotational inertia.

• The turning effect of a force applied at a point on a rigid body about the axis of rotation.
• Torque () is a pseudo vector directed along the axis of rotation. Its direction is given by the right hand fist rule as in the case with angular momentum. Unit of torque is N-m.
• It’s dimensional formula is
• If a body executes rotator motion, it is not necessary that a torque acts on it.
• Torque is necessary for producing angular acceleration only.

Work done by a force F, acting on a particle of a body rotating about a fixed axis: the particle describes a circular path with centre C on the axis : arc gives the displacement of the particle.
Let’s assume mass m is moving with constant velocity along a line parallel to the x-axis, away form the origin. It’s angular momentum wrt origin remains constant. L =mvy

Both theorems are related to moment of inertia.

Let us assume a small object of uniform density rolls up a curved surface with an initial velocity v. It reaches up to a maximum height of with respect to the initial position.
Now we need to find, which is this object?
In this scenario, Loss of Kinetic Energy = Gain in Potential Energy
; put given value of h in this equation.


This represents moment of inertia for Disc (with axis of rotation passing through C.G and perpendicular to its plane).

Body	Axis	Figure	I
Thin circular ring, radius R	Perpendicular to plane at centre		MR2
Thin circular ring, radius R	Diameter
Thin rod, length L	Perpendicular to rod, at mid point
Circular disc, radius R	Perpendicular to disc at centre
Circular disc, radius R	Diameter
Hollow cylinder, radius R	Axis of cylinder		MR2
Solid cylinder, radius R	Axis of cylinder
Solid sphere, radius R	Diameter

The moment of linear momentum of a given body about an axis of rotation is called as its angular momentum. If p be the linear momentum of a particle and r is its position vector from the point of rotation then .
When a body is projected at an angle with the horizontal in the uniform gravitational field of the earth, the angular momentum of the body about the point of projection increases, as it proceeds along the path.
Conservation of angular momentum and its applications: According to the law of conservation of angular momentum, if no external unbalanced torque is acting on system , then total vector sum of angular momentum of different particles of the system remains constant.
If no external torque acts on a body, the angular momentum of the body remains constant.

• Every point on a rotating body has an angular velocity and a linear velocity.
• Objects farther from the axis of rotation will move faster.
• The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation..
• The moment of inertial of a cube about it’s body diagonal is minimum.
• During pure rotational motion, the axis of rotation passes through centre of mass.
• Axis of rotation is always perpendicular to plane of rotation.
• If radius of gyration of the body is equal to its radius, then the body is a ring.
• Theorem of perpendicular axis is applicable only to thin lamina like sheet, disc, ring, etc. While parallel axes theorem is applicable to all type of bodies.
• Total K.E. is maximum for ring and minimum for solid sphere for same mass and moving with same speed.
• If metallic disc is melted and is moulded in the form of a sphere, then M.I. will decrease.
• Angular momentum is an axial vector, as it points along the axis of rotation.
• If there had been a single propeller in a helicopter, then due to non-conservation of angular momentum, the helicopter would turn in the reverse direction.
• When a person standing on a rotating platform raises his arms, M.I. of the system increases. Hence ω decreases.
• Angular Impulse, is the product of torque and time for which the torque acts.
• Axis of rotation of the rolling body is parallel to the plane on which it rolls.
• Acceleration of a rolling body does not depend on its mass.
• Theorem of perpendicular axis is applicable only to thin lamina like sheet, disc, ring etc. whose thickness is very small as compared to other dimensions (length, breadth or radius).
• Graph between M.I. (I) and Radius of Gyration (K) is parabola [I=MK²]
• Graph between Rotational K.E. (E_R) and angular velocity (ω) is a parabola .
• Graph between Angular momentum (L) and angular velocity is a straight line. L = Iω
• Graph between Angular momentum (L) and K.E. of rotation (E) will be parabola.