## Desk Chair Angular Momentum

posted on 25 Mar 2014 by guy
last changed 28 May 2015

 4.5 Average: 4.5 (2 votes) Select ratinghated itdidn't like itliked itreally liked itloved itCancel rating Your vote (click to rate)ages: 6 to 99 yrs budget: $0.00 to$0.00prep time: 0 to 0 min class time: 10 to 30 min Describes a simple demonstration of angular momentum conservation using a spinning desk chair. A short tutorial of angular momentum and angular momentum conservation is included.
subjects: Astronomy, Engineering, Physics

Find a desk chair that spins. Sit in it, hold your arms straight out to the sides, stick your legs straight out in front of you, and have a friend give you a gentle spin. While you're spinning slowly, pull in your legs and arms. You'll speed up.

Stick out your arms and legs again; you'll slow down. Pull them in again; you'll speed up. If you're desk chair doesn't have too much friction you may have time to pull your limbs in and out several times before you slow to a stop.

If you want to make the effect more dramatic, strap on some ankle weights and hold something heavy in your hands. A couple of 500-page physics texts would be appropriate.

Challenge your friends to see who can make the most rotations before slowing to a stop.

## what's going on?

The spinning desk chair provides an illustration of the "conservation of angular momentum", which is one of the fundamental conservation laws in physics and closely mirrors the conservation of linear momentum, which is more familiar to most people.

The conservation of linear momentum states that the mass times velocity, $mv$, of an object (or system of objects) does not change unless there is some external force acting on it. The principle follows from Newton's first law, which says that objects in motion will continue in a straight line at the same speed ("uniform motion") unless acted on by an outside force. If you leave it alone, an object will continue to move the same way it has, at a constant speed, not changing direction.

Similarly, the conservation of angular momentum states that a rotating object will continue rotating (in a way we'll specify below) unless acted on by a "torque". A torque is produced by a force that causes something to spin (or stop spinning).1 For a small object moving in a circle, any force on the object that has a component perpendicular to the radius (the line from the center of the circle to the object) produces a torque; it causes the object to rotate faster or slower. See figure 1 for an example of the forces on a roulette ball. The force from the rail is directed radially and therefore provides no torque (and does not change the angular momentum of the ball). The force of friction is perpendicular to the radius, providing a torque, which reduces the angular momentum.

### definition of angular momentum

In the absence of a net force, the linear momentum does not change;2 we say it is a "conserved quantity". In the absence of a net torque, which means there is no net force perpendicular to the radius of motion (although there could be forces along the radius), another quantity is conserved. In this case, it is tempting to guess that the conserved quantity would be the mass times the velocity that's perpendicular to the radius, $mv_\perp$. However, as the desk chair experiment shows, this guess is not correct. When you pull your arms and legs inward, you exert forces along the radius, but you exert no force perpendicular to the radius (no torque). Nonetheless, $v_\perp$ increases and you start rotating faster, $mv_\perp$ is not constant.

It turns out the conserved quantity is the radius, times the mass, times the perpendicular velocity, $rmv_\perp$, which is known as the angular momentum. As the radius decreases and the object is pulled inward, the perpendicular velocity $v_\perp$ increases, and vice versa.

## examples

Ice skaters use the conservation of angular momentum to great advantage when doing spins. A skater will start out spinning slowly with her limbs extended, but as she pulls them in towards her body, the average radius decreases and the rotation speed increases. Notice how the rotation speed changes as Natasha Lipinsky does her spins at http://www.youtube.com/watch?v=Mn2DRYvEODQ.

Celestial bodies that orbit planets or stars also conserve angular momentum. In these cases, the forces (due to gravity) are always directed inward towards the planet or star that is doing the pulling. There is no torque. Comets are particulary dramatic examples of the conservation of angular momentum because their orbits are so eccentric. The animation by Matthias Kabel at http://upload.wikimedia.org/wikipedia/commons/2/27/Komet.ogg shows how the speed of a comet increases as it gets closer to the sun.

• 1. The torque can be defined as the force on an object that is perpendicular to the radius, times the length of the radius, $rF_\perp$.
• 2. If linear momentum is conserved, and the mass is not changing, then the velocity is also constant.