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Angular Momentum Torque

The position r and momentum p = m v of a moving particle is defined with respect to a fixed reference point (origin) O in a given inertial frame. The moment of momentum or angular momentum L of the particle about O is defined by,

L = r × m v = r × p

Differentiating L, we find,

Since . The quantity is called the moment of (external) force or torque about the point O. The above relation states that the rate of change of angular momentum of a particle is equal to the net external torque acting on it, about a fixed origin O.

Hence, if , L is a constant of motion about O. This is known as the principle of conservation of angular momentum.

Note that the actual value of L depends on the particular choice of reference point O from which the position vector r (and hence momentum p = m dr/dt) of the particle is measured. Hence, the physical significance of L depends upon the significance of the choice of O. There are two special cases in which O has an inherent physical importance. First, when O represents the origin of the force acting on the particle, as for example in a central force problem. In case of an electron moving around a nucleus, or a satellite moving around earth, the nucleus (or earth) represents the origin of force about which the measurement nucleus (or earth) represents the origin of force about which the measurement of angular momentum of electron (or satellite) becomes a quantity of physical importance. The second case is that of an extended system, where centre of mass forms a physically meaningful point of reference. The angular momentum of an extended system about its center-of-mass is also referred to as ‘intrinsic’ angular momentum of the system. It plays an important role in the dunamics of rigid bodies.