For example: say you have two pendulums side by side that drop .051 meters, both pendulum bobs are moving 1 m/sec at the bottom. Suppose you devise a way of giving all the motion to one of the two bobs, the bob in motion will not rise .102 meters, it will rise .2038 m. You have doubled the energy. d = 1/2v?/a

This analogy is wrong. Yes in a perfect enviroment the bob in motion would reach .2038m...double the original height. However there is only ONE bob now at the high point, ie HALF the original mass so the potential energy at this point is EXACTLY the same as the kinetic energy at the low point when one of the bobs magically passes its energy to the other one. End result is ZERO change on total energy.

Hans and @ everyone,

the kid on the swing, will have a real hard time getting any momentum to accelerate, if he/she doesn't have arms and hands, that effectively, with energy, shorten the distance from the fulcrum.  thus providing additional accelration that coresponds to more height.  not applicable.  just jmho.

lol
sam

Pasted from an elementary physics site:

(http://www.hk-phy.org/articles/swing/swing_cm.gif)

Let's talk about playing swing again. As the child grows up, he won't be satisfied with the push and will want to learn how to play swing by himself. Please refer to Fig. 2, the broken line represents the route of the centre of mass of the child. Why can he swing with an increasing amplitude? When the boy swings to point A, he rises his body. Assuming that he rises his centre of mass by a distance  and the tension of the rope is , he does a positive work which supplies an energy of into the system. When the boy swings to point B, he letdowns his body to lower his centre of mass. This time he does a negative work. If the tension of the rope is , then the system loses  of energy. When the boy is at point B, his velocity is zero, so  is just the vertical component of his weight (, see Fig. 3). But when he is at point A, his velocity is the greatest, and the difference between the tension and his weight equals the centripetal force (). Hence we conclude that . Therefore, his positive work is greater than the negative one and energy is fed to the swing. If he supplies energy rhythmically, just like his father's push, resonance is produced and the boy swings with an increasing amplitude.

Hans von Lieven

The kid inputs energy into the 'pendulum' swing through his arms and bum.
Have you ever seen a swing with rigid supports instead of flexible rope or chain?.
No, because that would eliminate the energy transfer mechanism.

Not true Roy, this works with rigid arms as well.

Hans von Lieven

(http://www.johnlewis.com/jl_assets/product/230176252.jpg)

The swings on both ends have rigid arms, no ropes, no chains.

Hans von Lieven

When a kid leans back on a swing, they go higher and higher. They're increasing their acceleration.
And if a weight attached to the pendulum is doing the same thing, why would it be different ?

Kids on swings would seem to be very clever...  It's not about leaning back and forth.

They lean away from the swing's pivot point near the top of their swing, and pull themselves back up towards the pivot near the bottom.  When they pull back up, they work against more gravity and centrifugal force than are acting on them when they lean away, because they are close to the bottommost, fastest part of their swing.  The extra work they do there adds kinetic energy to their swinging motion.

The interesting thing is that kids don't know this, or need to know it.  If you're squirming around on a swing, there's really nowhere that any work you do can go, except into your swinging motion, so all you have to do is find something that's hard to do, and it will make you swing faster.  If kids practice swinging for a while, their bodies learn this all by themselves.  They learn to lean up, raise their feet, and bend the chains to raise their bottoms, because they can tell when they're working hard and they intuitively come to understand that work -- any kind of work -- produces speed.

In other words, kids on swings understand the conservation of energy :-)

Cheers,

Mr. Entropy

Sorry Roy,

even if they sit on a rigid bar welded to the arms it will still work, a little uncomfortable perhaps, but it will still work.

Hans von Lieven

Roy: two pendulum bobs that swing down from .051 m have a velocity of 1 m/sec. If these two bobs are on the opposites sides (and moving the same direction around the circle) of a horizontal circle they can be used as a cylinder and spheres apparatus. As the spheres swing out on the end of tethers all the motion is transferred to a small portion of the mass.

d = 1/2v?/a; if velocity is 1 m/sec rise (d) is .051 m, if velocity is 2 m/sec rise is .2038 m. (a) is acceleration 9.81 m/sec  d = 1/2v?/a   or   Ã¢Ë†Å¡2ad = v

If by any means you can transfer the momentum of a massive object to a smaller portion of itself you have made energy. The cylinder and spheres is one means and the rolling ring is probably another.