Torque and Angular Momentum for a system of Particles

Rajesh Naik

To get the total angular momentum of a system of particles about a given point we need to add vectorially the angular momenta of individual particles. Thus, for a system of n particles,

The angular momentum of the i^th particle is given by l_i = r_i × p_i where r_iis the position vector of the i^th particle with respect to a given origin and p = (m_i v_i) is the linear momentum of the particle. (The particle has mass m_iand velocity v_i ). We may write the total angular momentum of a system of particles as

……………………………………………………………………(25b)

This generalization of the definition of angular momentum (Eq. 25a) for a single particle to a system of particles.

Using Eqs. (23) and (25b), we get

…………………………………………………..(28a)

where τ_i is the torque on the i^th particle:
τ_i = r_i × F_i

The force F_ion the i^th particle is the vector sum of external forces F_i^ext acting on the particle and the internal forces F_i^int exerted on it by the other particles of the system. We may therefore separate the contribution of the external and the internal forces to the torque

We shall assume not only Newton’s third law. i.e the forces between any two particles of the system are equal and opposite, but also that these forces are directed along the line joining the two particles. In this case the contribution of the internal forces to the total torque on the system is zero, since the toque resulting from each action-reaction pair of forces is zero. We thus have τ_int =0 and therefore τ = τ_ext.

Since r = ∑τ_iit follows from Eq. (28a) that

dL/dt = τ_ext …………………………………………………………………………..(28b)

Thus the time rate of the total angular momentum acting on the system taken about the same point Eq. (28b) is the generalization of the single particle case of Eq. (23) to a system of particles. Note that when we have only one particle, there are no internal forces or torques. Eq. (28b) is the rotational analogue of

dP/dt = F_ext …………………………………………………………………………..(17)

Note that like Eq. (17), Eq. (28b) holds good for any system of particles, whether it is a rigid body or its individual particles have all kind of internal motion.