Law Of Conservation Of Angular Momentum

Introduction to the Law of Conservation of Angular Momentum:

As we have conservation of linear momentum, similarly we have its rotational counterpart, namely, the law of conservation of angualar momentum. In this article we shall touch some of its aspects

Angular momentum of a particle about a point 'O' is defined as 'vecL = vecr xx vecp' where 'vecL' is the angular momentum of the particle about the point 'O', 'vecp' is the linear momentum and 'vecr' is the position vector of the particle from the given point 'O'. The angular momentum of a system of particles is the vector sum of the angular momenta of the individual particles.

The magnitude of the angular momentum of a particle of mass m moving with velocity v is

L = rp sin'theta'

= r(mv) sin'theta'

= mv(rsin)'theta'

=( linear momentum)(Perpendicular distance of line of motion from the point)

If the line of motion is perpendicular i.e., 'theta' = 900 , L = m v r .

Angular momentum is also called moment of momentum.

Statement of Law of Conservation of Angular Momentum

Statement : If there is no resultant external torque on a rotating system, the angular momentum of the system remains constant both in magnitude and direction.

Proof : The resultant external 'tau' acting on a rotating system is related to its angular momentum 'vecL' as

'vec(tau)' = '(vec(dL) / dt)'

If the resultant external torque, 'vectau' is equal to zero,

'vec(dL)/(dt)' = 0.

As the differentiation of a constant quantity is zero, 'vecL ' is constant. Hence , the angular momentum of a rotating system is constant when there is no resultant torque.

Examples of Law of Conservation of Angular Momentum

1. A man stands on a turn table with dumb-bells in his stretched hands. The turn table is set into rotation at a constant angular velocity 'omega' . If the man now starts to bring his hands closer to his body we will observe that the angular velocity increases gradually. The angular velocity becomes maximum when the man folds his hands. This can be explained using the principle of conservation of angular momentum. According to this principle, if there is no resultant external torque, L = constant i.e., I'omega' = constant. Angular velocity is inversely proportional to the moment of inertia. As the man brings his hands closer to his body I I decreases. so 'omega' increases.

2. A ballet dancer decreases or increases his angular speed of rotation by stretching the hands or bringing the hands closer to the body.

by: nayaknandan

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