When a rigid body rotates about an axis, it
possesses kinetic energy. All the kinetic energy of a rotating body is
rotational kinetic energy (Kr). Let us now find the kinetic energy of a
rotating body.
The particle of mass m1 follows a
circular path of radius r1. The magnitude of the linear or tangential velocity of the
particle on this circle is v₁.
Rotational kinetic
energy of the particle = ½ m₁ v₁² = ½
m₁ (r₁ ω)² = ½ m₁ r₁² ω²
Similarly, the rotational kinetic energy of particles of
masses m₂,
m₃ ...... are ½ m₂
r² ω², m₃ r² ω²… respectively.
The rotational kinetic energy Kr of the
body s equal to the sum of the rotational kinetic energies of all the
particles.
= ½ (m₁
r₁² + m₂ r₂² + m₃ r₃² + …..) ω²
= ½ (∑mᵢ
rᵢ)² ω²
But ∑mᵢ rᵢ² is the moment of inertia I of the body about the given axis
of rotation.
Rotational K.E. of the
body, Kr = I ω²
Thus the rotational
K.E of a body is equal to half the product of the moment of inertia of the body
and the square of the angular velocity of the body about the given axis of
rotation, Note that once again we could have guessed its general form. In
analogy to ½ m v², we see that v is replaced by ω and m by I.
For continuous mass
distribution, the moment of inertia, I, is given by
I = ∫r² dm
Where r is the
distance of the mass-element dm from the axis of rotation. The integration is carried out over the
entire mass distribution.
Moment of inertia of a
system of particles depends on
a. Axis of rotation
b. Mass of the system
c. Distribution of mass in the body
Moment of inertia plays same role in
rotational motion as mass plays in translational motion. It is the property of
the body by which body opposes any change in its state of rotational motion.
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