Angular Momentum
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Angular Momentum of a Particle: |
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The angular momentum of an object rotating about some reference point is the measure of the extent to which the object will continue to rotate about that point unless acted upon by an external torque. In particular, if a point mass rotates about an axis, then the angular momentum with respect to a point on the axis is related to the mass of the object, the velocity and the distance of the mass to the axis.
The angular momentum of a
particle (L) relative to the origin O is defined as the cross product of the particle’s instantaneous position vector (r)
and its instantaneous linear momentum (p): L = r × p.
The SI unit of angular momentum is kg.m2/s. Both
magnitude and direction of L depends upon the choice of origin.
Now, the angular momentum can be given
by,
L = mvr(sin¢); where ¢ is the angle between
r and p. When r is parallel to
p, L is zero. When the linear velocity
of a particle is along a line passing through the origin, the
particle has zero angular momentum with respect to the
origin. If r is perpendicular to p then
L=mvr. At that instant,
the particle moves exactly as if it were on the rim of a wheel rotating about the origin in a plane defined by r
and p.
Now the net torque acting on the particle is given by,
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Thus, the net torque acting on a particle is equal to the time of change of the particle’s angular momentum. It is important to note that same origin must be chosen for both torque and angular momentum.
Angular momentum for a system of Particles:
where the vector sum is over all n particles in the system. The total angular momentum can vary with time only if a net external torque is acting on the system so that we have
Thus, the time rate of change of the total angular momentum of a system about some origin in an inertial frame is equals to the net external torque acting on the system about that origin.
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Newtonian Mechanics