# Why is angular momentum conserved but not linear?

Jul 1, 2014

Angular and linear momentum are not directly related, however, both are conserved. Angular momentum is a measure of an object's tendency to continue rotating. A rotating object will continue to spin on an axis if it is free from any external torque.

Linear momentum is an object's tendency to continue in one direction. An object traveling in a given direction with a certain velocity will continue to do so until acted on by an external force (Newton's 1st law of motion). Since angular and linear momentum both have magnitudes and directions associated with them, they are both vector quantities.

Angular momentum is given by:

$L = r \cdot m {v}_{\text{tangential}}$

where $L$ is the angular momentum, $r$ is the radius of the mass relative to the axis of rotation, m is the mass of the object, and ${v}_{\text{tangential}}$ is the velocity vector of the mass tangent to the radius of rotation

Linear momentum is given by:

$p = m v$

where $p$ is the linear momentum of the object, $m$ is the mass, and $v$ is the velocity of the object in the direction of travel

A common example of the conservation of angular momentum in the physics classroom is spinning in a chair with weights in each hand. When the weights are brought in, the student will rotate faster (because the radius is smaller). The opposite is also true. When the student extends his hands, the chair will slow down.

A common example of the conservation of linear momentum can be seen in collision mechanics. In a perfectly inelastic collision, if two objects of the same mass collide, and one starts at rest, the final velocity of the system will be exactly 1/2 of the velocity of the mass that was moving originally:

${p}_{m} 1 + {p}_{m} 2 = {p}_{m} 1 m 2$
${m}_{1} {v}_{1} + {m}_{2} {v}_{2} = \left({m}_{1} + {m}_{2}\right) {v}_{12}$

Since ${m}_{2}$ started at rest, ${v}_{2} = 0$, and we are assuming ${m}_{1} = {m}_{2}$:

${m}_{1} {v}_{1} = 2 {m}_{1} {v}_{12}$
${v}_{12} = \frac{1}{2} {v}_{1}$