Angular momentum of a particle :
The angular momentum in a rotational motion is similar to the linear momentum in translatory motion. The linear momentum of a
particle moving along a straight line is the product of its mass and linear velocity (i.e) Vec p = m.

*Angular
momentum of a particle*

The angular momentum in a
rotational motion is similar to the linear momentum in translatory motion. The
linear momentum of a

particle moving along a
straight line is the product of its mass and linear velocity (i.e) Vec p = m.
Vec v. The angular momentum of a particle is

defined as the moment of
linear momentum of the particle.

Let us consider a particle of
mass m moving in the XY plane with a velocity v and linear momentum p = mv at a distance r from the origin (Fig. ).

The angular momentum L of the
particle about an axis passing through O perpendicular to XY plane is defined
as the cross product of Vec r and Vec p.

(i.e) Vec
L = Vec r ? Vec P

Its magnitude is given by *L =
r p sin* *θ*

Where θ is the angle between Vec *r*
and Vec *p* and L is along a direction perpendicular to the plane
containing Vec *r *and
Vec *p*
.

The unit of angular momentum is kg m^{2} s^{?1} and
its dimensional formula is, M L^{2} T^{?1}.

*Angular momentum of a rigid body*

Let us consider a system of n particles of masses *m*_{1}, *m*_{2} ?.. *m*_{n}
situated at distances *r*_{1}, *r*_{2}, ?..*r*_{n} respectively from the axis of rotation (Fig. ).
Let *v*_{1},*v*_{2}, *v* _{3} ?.. be the linear velocities of the particles
respectively, then linear momentum of first particle = *m*_{1}*v*_{1}.

Since v_{1}= r_{1}ω the linear
momentum of first particle = m_{1}(r_{1} ω)

The moment of linear momentum of first particle

= linear momentum ? perpendicular distance

= (m_{1}r_{1}ω) ? r_{1}

angular momentum of first particle = m_{1}r_{1}^{2}ω

Similarly,

angular momentum of second particle = *m*_{2}*r*_{2}^{2}ω

angular momentum of third particle = *m*_{3}*r*_{3}^{2}ω and so on.

*The sum
of the moment of the linear momenta of all the particles of a rotating rigid
body taken together about the axis of rotation is known as angular momentum of
the rigid body.*

Angular momentum of the rotating rigid body =
sum of the angular momenta of all the particles.

(i.e) L = *m _{1}r_{1}*

L = ω [ *m _{1}r_{1}*

∴
L = ωI

where I = ∑ *m _{i} r_{i}*

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