Translational momentum is commonly called linear momentum and is based on the linear motion of a system of particles, such as those particles that make up a rigid or deforming body. There is no restriction on the way the particles are connected when we speak of linear momentum.

Rotational momentum is commonly called angular momentum and is based on the rotation of a system of particles about a given point. It is common to associate angular momentum with bodies that may either be rigid or deforming. However, the discussion here will focus on the angular momentum of rigid bodies.

Conservation of momentum, for linear and angular (rotational) motion, will be discussed here in detail.

The translational motion of a system of particles that experiences no external linear impulse can be analyzed using conservation of linear momentum (which is, conservation of momentum applied to linear motion).

From the page on linear momentum we derived the following vector equation for a system of particles:

Where:

The term

is defined as the external linear impulse acting on the system of particles (between time

Linear momentum is conserved if no external forces act on the system of particles. This means that:

Equation (1) therefore becomes

Mass cancels out and

This is a nice result for conservation of momentum for the linear case. This result tells us that the velocity of the center of mass does not change if no external forces act on the system of particles. This is true even if there is collision between the particles. Collisions result in internal forces that occur in equal and opposite pairs (Newton’s Third Law). Hence, when summing over all the particles, the internal forces cancel each other out.

Equation (2) can be written as:

Where:

Equation (3) is perhaps the most useful mathematical expression for the conservation of momentum (for the linear case) of a system of particles. It can be used to solve problems involving elastic collisions or inelastic collisions between bodies, which can be treated as particles, and which are subject to no external forces (such as friction on a surface). To see an example of a solved problem that makes use of conservation of momentum for the linear case, go to The Physics Of Billiards.

The rotational motion of a body that experiences no external angular impulse can be analyzed using conservation of angular momentum (which is, conservation of momentum applied to angular motion). The analysis that follows will be for a rigid body.

From the page on angular momentum we derived the following scalar equation for a rigid body experiencing planar (two-dimensional) motion:

Where:

The term

is defined as the external angular impulse acting on the rigid body (between time

Angular momentum is conserved if no external moments act on the rigid body. This means that:

taken about either point

Equation (4) therefore becomes

As a result, angular momentum is conserved for the rigid body (between

Let's now extend conservation of momentum for the angular case to three-dimensional motion. From the page on angular momentum we derived the following vector equation for a rigid body experiencing general three-dimensional motion:

Where:

The term

is defined as the external angular impulse acting on the rigid body (between time

Angular momentum is conserved if no external moments act on the rigid body. This means that:

taken about either point

Equation (5) therefore becomes

As a result, angular momentum is conserved for the rigid body (between

To analyze the motion of a system (or body) when there is no external impulse acting on the system (or body), one can apply the equations of motion, such as those given in the rigid body dynamics page. So for the case of angular motion with no external moments, you just set the moments equal to zero in the equations of motion, and the solution will correspond to the case where angular momentum is conserved.

If you want to see an interesting paper explaining the conservation of angular momentum for semi-rigid bodies experiencing three-dimensional motion, have a look at:

See also, Cat righting reflex and physics of gymnastics.

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