**Kinetic energy of a rotation body**

Consider a rigid body of mass M rotating about an axis YY' passing through it with angular velocity ω made of n particles of mass m

_{1}, m_{2}, m_{3}, …, mn at distance r_{1}, r_{2}, r_{3}, …, r_{n}from the axis. Although the angular velocity is same for all the particles but the linear velocity won’t be same for all the particles. So, let v_{1},v_{2}, v_{3}, ……., v_{n}be the linear velocities of the particles.Then, K.E. for m

_{1}= ½ m_{1}v_{1}^{2}………. (i)We know, v = r. ω

Then, v

_{1}= r_{1}. ωEquation (i) becomes, K.E. for m

_{1}= ½ m_{1}. R_{1}^{2}. ω^{2}For m

_{1}, m_{2}, m_{3}, ……., m_{n}K.E. for m

_{2}= ½ m_{2}. R_{2}^{2}. ω^{2}K.E. for m3 = ½ m

_{3}. R_{3}^{2}. ω^{2}K.E. for mn = ½ m

_{n}. R_{n}^{2}. ω^{2}For total kinetic energy,

K.E.

_{rot}= ½ m_{1}. R_{1}^{2}. ω^{2}+ ½ m_{2}. R_{2}^{2}. ω^{2}+ ½ m_{3}. R_{3}^{2}. ω^{2}+ ……. + ½ m_{n}. R_{n}^{2}. ω^{2}K.E.

_{rot}= ½ ω^{2}(m_{1}. R_{1}^{2}+ m_{2}. R_{2}^{2}+ m_{3}. R_{3}^{2}+ ……. + m_{n}. R_{n}^{2})K.E.

_{rot}= ½ Iω^{2}[.: m_{1}. R_{1}^{2}+ m_{2}. R_{2}^{2}+ m_{3}. R_{3}^{2}+ ……. + m_{n}. R_{n}^{2}= I]
## Share on Social Media