Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Rotational Motion, Angular Momentum, and Dynamics of Rigid Bodies

Michael Cohen

Recall that an inertial frame is a set of axes such that, if you measure positions and velocities with respect to those axes, Newton’s first law is true; that is, a particle subject to no forces will move with constant velocity. In particular, the axes of an inertial frame must be nonrotating with respect to the background of the distant stars. Regarding the motion of the origin of an inertial frame, there is some arbitrariness due to imprecision in the notion of “no force.” We shall assume here that we understand the meaning of “inertial frame” well enough to solve elementary problems.

Consider a particle of mass whose position vector with respect to the origin of an inertial frame is and whose velocity and acceleration are and . Taking the cross-product of both sides of the equation of motion with we obtain

where is the total force acting on the particle. The left side of (1) is, of course, the torque (about the origin ) acting on the particle. We also define the angular momentum of the particle about the origin by the equation

Using the rule for differentiating a cross-product we find

Since we can combine (1) and (3) to obtain

In words, the torque is equal to the rate of change of angular momentum (similar to the statement that the force is equal to the rate of change of linear momentum).

1. Angular Momentum and Central Forces

Equation (4) has important consequences when applied to the central force problem, that is, a particle moving under the influence of a force which is always directed toward a fixed point. If we take our origin at the fixed point, then the torque vanishes because and are parallel (or antiparallel). Therefore and the angular momentum is constant. The constancy of implies

  1. the motion of the particle takes place in a fixed plane, namely, the plane containing the force center, the initial position of the particle, and the initial velocity vector of the particle;
  2. as the particle moves in this plane, the vector from the force center to the particle sweeps out area at a constant rate (this is Kepler’s Second Law and is a property of all central forces, not just the inverse-square law).

To prove (a), we pass a plane through the force center , perpendicular to the constant vector . Equation (2) implies that is perpendicular to ; therefore lies in this plane. But since (where and are the initial position and velocity vectors), the plane perpendicular to is the plane containing and .

Figure 1: Direction of .