Kinetic Energy of Rotation , Moment of Inertia

K.E of Rotation & Moment of Inertia :

A rigid body undergoes pure rotational motion about a fixed axis n and its constituent particles move on circular paths with radii r, r, … and rn (say) with linear velocities v1 = ωr, v2 = ωr2 , ….. and vn= ωr, ω being the angular velocity of the rigid body.

If m1, m2, ……… and mn are the masses of the respective particles of the rigid body, then the kinetic energy of the system is given by

$\displaystyle K. E = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 + \frac{1}{2}m_3v_3^2 + …. + \frac{1}{2}m_nv_n^2$

$\displaystyle K. E = \frac{1}{2}m_1\omega^2r_1^2 + \frac{1}{2}m_2\omega^2r_2^2 + \frac{1}{2}m_3\omega^2r_3^2 + …. + \frac{1}{2}m_n\omega^2r_n^2$

$\displaystyle K. E = \frac{1}{2}(m_1 r_1^2 + m_2 r_2^2 + m_3 r_3^2 + ….+ m_n r_n^2)\omega^2$

$\displaystyle K. E = \frac{1}{2}I\omega^2$

The term I = m1r12 + m2r22 + ………. mnrn2 is called Rotational inertia or Moment of inertia of the body or system of particles.

Rotational inertia of a particle of mass ‘ m ‘ is given by the following expression

I = mr2 ;

where r = perpendicular distance of the particle from the axis of rotation.

For a continuous distribution,

$\displaystyle I = \int r^2 dm$

where dm is a small element of the body at a distance r, from the axis of rotation.

The kinetic energy of a rigid body, due to rotation, is given by

$\displaystyle K. E = \frac{1}{2}I\omega^2$

Moment of inertia of a system of particles depends on :

(i) Axis of rotation

(ii) Mass of the system

(iii) Distribution of mass in the body

The moment of inertia of a rigid body about a given axis of rotation, is a constant.
Moment of inertia plays the same role in rotational motion as mass plays in translational motion.