The Swing...?

Question

Pull a swing or pendulum to one side, let it go and it will swing back and forth by itself. As it goes back and forth the swing is conserving:

a) angular and linear momentum
b) only angular momentum
c) only linear momentum
d) neither angular momentum nor linear momentum

The answer is: d. First the pendulum swings to the right, next it moves back to the left. If the momentum going to the right is +5, then when it swings back the momentum must be -5, and when it stops to come back at the end of the swing, the momentum is zero. So the momentum changes from +5 to zero to -5 to zero to +5 to .... It is always changing and so it can't be conserved. What about angular momentum? Well, first it pivots clockwise about the suspension point and at the end of a swing it is momentarily not pivoting at all, and then it pivots counterclockwise, repeating the cycle over and over. So angular momentum is not conserved either. How come? Perhaps you thought momentum was always conserved? Where does the momentum go? When a ball hits a bat where does the momentum go? Into the bat. The linear momentum of the pendulum goes into the string, then into the ceiling, and then into the earth. The angular momentum of the pendulum goes directly into the earth by gravity.

That means the earth gets anything the pendulum loses. As the pendulum moves to the right, the earth moves a tiny tiny tiny bit to the left. As the pendulum turns clockwise the whole earth turns a tiny tiny tiny bit anti-clockwise and of course when the pendulum swings the other way the whole earth also reverses its tiny motion. Suppose the earth's mass is a billion times larger than the mass of the pendulum and the pendulum moves one foot to the right. How far does the whole earth move? One billionth part of one foot to the left.

Answer 1

A. The pendulum itself has variable angular and linear momentum, but these are conserved when the earth-pendulum system is considered. Can the pendulum alone be said to conserve or not conserve momentum? That's more a semantic than mechanical question. I'd say no; the idea of conservation can only apply to the whole system.
Edit: It's similar to the problem of a ball thrown against a wall and bouncing back. Clearly the momentum of the ball, considered as a "system", is not conserved. Momentum is a vector, and the direction change of the bounce means the momentum is different, even if it retains the initial speed. But fear not, we are told, the system is more than the ball; the wall, the earth, etc. are also in the system, they respond infinitesimally to the bounce force, and with no force external to the system, the total momentum is unchanged and the eternal law stands supreme.
So, what are the specific mechanics of conservation in the pendulum case? Somehow, an oscillating force and torque must be getting transmitted to the earth, and this must occur through the suspension. It's easy to envision the force transmission; that's what a pivot or a string does. But how is the torque transmitted? The force itself is almost always applying a torque to the earth, since its line of action only passes through the earth's center of mass at bottom dead center. So although you can't measure a torque at the pivot or string attach point, the earth sees one nonetheless. Each one-way pass of the pendulum works like a reaction wheel on a spacecraft. Spin up the wheel and the spacecraft will change its attitude as long as the wheel keeps spinning; stop the wheel and the spacecraft stops too.

Answer 2

There is so much more to it than how long have you been together. Swinging can either be a good thing or a very bad thing... the deciding factor is how strong is your relationship. Is your communication good enough that you BOTH feel you can talk to each other about anything and everything? Do you both trust each other completely? Are there any jealousy issues? If there is any lack of trust or jealousy issues, don't do it. Work those issues out first. If not and you can say without a doubt that your communication is A1 and swinging is something that you both want to do and you can agree on what your goals are as they pertain to swinging and what your rules will be, then go for it and have fun.

Answer 3

Answer is A, Angular and Linear Momentum.

- Angular Momentum due to the angle of the arc the pendulum traverses.
- Linear Momentum due to the tendency of the mass at the end of the pendulum to traverse a straight (linear) line.