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MOMENTUM AND ANGULAR MOMENTUM (OR MOMENT OF MOMENTUM)

5.9.1. Momentum. The product of mass and velocity of a body is known as momentum of the body. If ‘m’ is the mass of a body and v is the velocity of the body, the momentum of the body is equal to m × v.

5.9.2. Moment of Momentum or Angular Momentum. The product of mass moment of inertia and angular velocity of a rotating body is known as moment of momentum or angular momentum.

If ω = Angular velocity of a body rotating about an axis, and

I = Moment of inertia of the body about the axis.

Then angular momentum = ω × I                                               …(5.35)

The equation (5.35) is derived as given below:

Consider a body of mass ‘m’ rotating in a circle about its centre O.

Let dm = Mass of the elementary strip

r = Radius of the mass ‘dm’

ω = Angular velocity of the body or angular velocity of the mass ‘dm’, (As angular velocity is constant, hence the angular velocity of the body will be same as angular velocity of mass ‘dm’)   .

v = Linear velocity of mass ‘dm’ = to × r.

Now momentum of elementary mass

= Elementary mass × Velocity,

=dm × v

= dm × ωr2

Moment of momentum of elementary mass ‘dm’ about O

= Momentum × Radius

= (dm × ωr) × r

= dm × ωr2                                                                                                        … (i)

The moment of momentum of the entire mass about O is obtained by integrating the above equation.

Moment of momentum of the entire mass

MOMENTUM AND ANGULAR MOMENTUM (OR MOMENT OF MOMENTUM)