When a body moves along a straight line, then
the product of the, mass m and the linear velocity v of the body is called the
“linear momentum” p of the body (p = m * v), if a body is rotating about an
axis, then the sum of the moments of the linear momenta of all the particles
about the given axis is called the “angular momentum” of the body about that
axis. It is represented by ‘J’.
Let a body be
rotating about an axis with an angular velocity ω. All the particles of the
body will have the same angular velocity, but their linear velocities will be
different. Let a particle be at a distance r1 from the axis of
rotation. The linear velocity of this particle is given by
v1 = r1 ω
If the mass of the particle be m₁, then its
linear momentum = m1 v1.
The moment of this momentum about the axis of
rotational
= momentum * distance
= m1 v1 * r1
= m1 (r1 ω) * r1
= m1 r1² ω
Similarly, if the masses of other particles
be m2, m3, … and their respective distances from the axis
of rotation be r2, r3, … the moments of their linear
momenta about the axis of rotation will be m₂ r₂² ω, m3 r3²
ω, … respectively. The sum of the moments of linear momenta of all particles,
that is, the angular momentum of the body is given by
L = m1 r1² ω + m2 r2² ω
+ m3 r3² ω + …
= (m1 r1² ω + m2 r2² ω
+ m3 r3² ω + …) ω
= (∑mr²) ω.
But ∑mr² is the moment of interia I of the
body about the axis of rotation. Hence the angular momentum of the body about
the axis of rotation is
L = I * ω
The S.I. unit of angular momentum is kgm²s-1
of Js. Its dimensional formula is [ML²T-1].
It is clear from the above formula that just
as the linear momentum (m v) of a body is equal to the product of the mass of
the body and its linear velocity; in the same way the angular momentum (I ω) of
a body about an axis is equal to the product of the moment of interia of the
body and its angular velocity about that axis.
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