# Rotational Kinetic Energy | Rotational Kinetic Energy of Disc and Hoop

The energy of an object due to its rotation about an axis is called rotational kinetic energy.

**Rotational kinetic energy**

The **energy** of an object due to spinning about an axis is called rotational kinetic energy. Rotational kinetic energy is directly proportional to the rotational inertia and the square of the magnitude of the **angular velocity**.

**Rotational kinetic energy formula**

Consider an object spinning about an axis with constant angular velocity Ï‰. Every point of the body has some K.E due to **circular motion**. Â In order to calculate the total K.E of the spinning body, imagine that it is composed of tiny pieces of masses m_{1}, m_{2}â€¦. ..,m_{n},.

Consider a single piece of mass m_{i}, at a distance r_{i} from the axis of rotation moving in a circle. Then its speed is given by

V_{i}= r_{i}Ï‰

K.E= 1/2m_{i}v_{i}^{2}

=1/2m_{i} (r_{i}Ï‰)^{2} =1/2m_{I}r_{i}^{2}Ï‰^{2}

The rotational K.E of the whole body is the sum of the kinetic energies of all the pieces.

K.E_{rot} = 1/2 m_{i}r_{i}^{2}Ï‰^{2} +1/2m_{2}r_{2}^{2}Ï‰^{2} +1/2m^{3}r_{3}^{2}Ï‰^{2}

=1/2(m_{1}r_{1}^{2}+ m_{2}r_{2}^{2}+â€¦â€¦â€¦. ) Ï‰^{2}

I= m_{1}r_{1}^{2} +m_{2}r_{2}^{2}+â€¦â€¦

**Rotational kinetic energy uses**

Rotational K.E is practically used by flywheels, which are essential parts of many engines. Flywheel stores energy between **power**s stokes of the piston which is distributed over the full revolution of the crankshaft and rotation remains smooth.

**Rotational Kinetic Energy of Disc and Hoop**

Consider a disc and a hoop started moving down on an inclined plane of height ‘h’. They have rotational as well as translational motion. If there is no friction, then the total K.E of the disc or hoop on reaching the bottom of the inclined plane must be equal to its P.E at the top.

P.E_{ at top }= Rotational K.E _{at bottom} + Translational K.E _{at} _{bottom}

**Rotational Kinetic Energy of Disc**

P.E = K.E_{rot} +K.E_{tran}

K.E_{rot}=1/2lÏ‰^{2}

K.E_{rot}=1/2 (1/2mr^{2}Ï‰^{2})

=1/4mr^{2}Ï‰^{2}

=1/4mv^{2}

P.E=1/4mv^{2 }+1/2mv^{2}

mgh=mv^{2}+2mv^{2}/4

mgh=3/4mv^{2}

v^{2}= 3/4gh

v=âˆš3/4gh

**Rotational Kinetic Energy of hoop**

P.E = K.E_{rot} +K.E_{tran}

K.E_{rot}=1/2lÏ‰^{2}

K.E_{rot}=1/2mr^{2}Ï‰^{2}

=1/2mv^{2}

mgh= mv^{2}+mv^{2}/2

mgh =2 mv^{2}/2

v^{2}=gh

v=âˆšgh

This is the final **velocity **of a hoop on reaching the lower end of an inclined plane. It shows that the velocity of the disc will be greater than the hoop on reaching the bottom of the inclined plane.

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