Conservation of angular momentum
THEORY
Consider a system consisting of a single particle of mass mi that has
position, ri and velocity, vi. Relative to the origin, the particle has an
angular momentum given by,
Li  ri x (mi vi),
Differentiating this we obtain,
dLi/dt = ri x (mi ai) = ri x Fnet,i  net,i
In the last step we defined the net torque on the ith particle.
Next, consider a system of particles.
Newton’s second law for a system of particles is
 F ext = dP/dt ,
where is  F ext is the net external force acting on the system and P is the
linear momentum of the system.
For the same system of particles, Newton’s second law for rotation is
  ext = dL/dt ,
where is   ext is the net external torque acting on the system and L is the
angular momentum of the system.
The conservation of linear momentum occurs if  F ext = 0, for then
Pi = Pf .
The conservation of angular momentum occurs if   ext = 0, for then
Li = Lf .
For a rigid body rotating about a fixed axis of rotation every particle of the
rigid body is in rotation with a radius, ri, and with a speed of vi (= ri  ).
Therefore,
Angular momentum of rigid body = L =
 r x m v =  r x m r  = I
i
i
i
i
i i
where the sum gives the moment of inertia of the rigid body.
For such a rigid body with zero net external torque acting on it, the
conservation of angular momentum is written,
Ii i = If f .
Comments
 This new conservation law manifests itself in many everyday occurrences,
for example, Kepler’s second law of planetary motion, a rotating skater or in
a tornado.
 One uses this principle in problem solving by following the Conservation
Law Recipe.
EXAMPLES
[in class]