Angular Momentum Vector

Angular momentum is analogous to the principle of impulse and momentum. The angular momentum about a point (generally the origin, O) is

The distance vector, r, is from the point of rotation to the object. This cross product will produce a vector, H, that is perpendicular to both r and v, as shown in the diagrams at the left.

Most problems deal with the angular momentum for an object moving in the x-y plane (about the origin O). Then the angular momentum equation simplifies to

     H_o = m|r||v| sinθ e_z

where r is the position vector of the object and e_z is the unit vector in the z direction.

The relationship between the angular momentum of an object and the external moment applied to the object can be determined by examining the cross product of the position vector and Newton's Second Law, or

     r × ΣF = r × ma = r × m dv/dt

The right-hand side of the this equation equals the time derivative of the angular momentum,

     d(r × mv)/dt = (dr/dt × mv) + (r × m dv/dt)

                       = (v × mv) + (r × m dv/dt)

                       = r × m dv/dt

Thus, the moment of external forces about O equals the object's rate of change of angular momentum about O:

Integrating this equation with respect to time gives

If the moment acting on an object about a fixed point O is known, it can be integrated to determine the object's change in angular momentum.

If the moment is zero, H_o is constant, i.e. angular momentum is conserved.

If the total force acting on the object remains directed towards a fixed point, such as the origin O, then the fixed point is called the center of motion, and the object is said to be in central-force motion.

The position vector r is always parallel to the force, thus the moment is zero:

     r × ΣF = 0 

Hence, for central-force motion, angular momentum is conserved, i.e. H_o is constant.

Central-Force Motion

If an object undergoes central-force motion in a plane, then it is said to be in plane central-force motion.

If r and v are expressed in cylindrical coordinates then

     r = re_r                     v = v_re_r + v_θe_θ

Substituting these into the equation for angular momentum gives

     H_o = r × mv

         = re_r × m (v_re_r + v_θe_θ)

         = mr v_θe_z

This equation indicates that for plane central-force motion, the product of the radial distance from the center of motion and the transverse component of velocity is constant, i.e. r v_θ is constant.