Over the Hill

1. Bend the steel shelf bracket to a profile similar to the one shown in the photos below, or design your own profile. (Important! Be careful to bend gently, and don't make sharp angles. Gentle curves work best. If you create a kink in the track, you may have to start over.) The hill section shown in our setup starts about 18 inches (45 cm) from the right end and is about 6 inches (15 cm) long; the hill itself is about 1/2 inch (13 mm) high.
   
2. Using the wood screws, attach the track to the base. Place the first screw through a hole in the track near the starting point; place the second screw a little before the first hill; and place the third screw at the bottom of the valley. Use the drill to create pilot holes if necessary.
3. Check the finished assembly to be sure it’s stable when used and that the ball can roll smoothly over all parts of the track.

Try to roll the marble so that it climbs from the flat end of the track over the small hill, but can’t get back again: It should stay trapped in the valley between the hill and the high end. Keep track of how many times it takes to get your first success and how many wins you get in your first ten tries.

Try another set of ten. Did your score improve? Is this a game of chance or a game of skill?

Where is mechanical work being done? Where is kinetic energy greatest? Where is gravitational potential energy greatest? Where is kinetic energy least? Where is gravitational potential energy least? Is conservation of energy being obeyed in this game?

This is a game of skill, not a game of chance. You will probably find that practice will significantly improve your ability. In the real carnival game, the participant pays for each try. On average, it’s likely that there will be enough unsuccessful tries, when compared to the number of prize-winning successful tries, for the game to turn a profit for carnival proprietors. (Of course, if the prize costs less than the price of a try, a profit is assured!)

This is an outstanding example of energy transformations involving work, kinetic energy, gravitational potential energy, and heat. You do mechanical work (exerting a force through a distance) on the marble to get it rolling on the flat track, and this work appears as kinetic energy. As the marble climbs up the small hill, kinetic energy is transformed to gravitational potential energy, and as it goes down the hill on the other side, this process is reversed. This transformation process is then repeated as the marble begins to climb the second hill, stops and reverses direction partway up, and then comes back down the hill. It is repeated yet again on the small hill on the way back. Along the entire journey, friction is constantly causing kinetic energy to be transformed to heat.

If friction causes enough kinetic energy to be transformed to heat after the marble initially gets over the small hill, then the marble won’t be able to get back to the top of that hill on its return. It will become trapped in the valley, losing more kinetic energy to heat each time it oscillates back and forth in the valley, until it finally comes to rest.

Kinetic energy is greatest immediately after you stop doing work to get the ball rolling. Maximum gravitational potential energy could occur at either the top of the first hill, or at the ball’s highest point on the second hill, depending on circumstances. Kinetic energy is least when the ball is stopped, which would be where it reverses direction on the second hill. Gravitational potential is least when the ball is at the lowest elevation, which would be on the initial flat section and at the bottom of the valley. Conservation of energy is definitely present in this device in that, at any given instant, the sum of the kinetic energy, potential energy, and heat are equal to the mechanical work initially done. Conservation of energy is expressed in the First Law of Thermodynamics.

Since the marble is rolling rather than sliding, it has both translational and rotational kinetic energy. In an advanced mathematical analysis, these could be considered separately.

If the device were 100% efficient—that is, if there were no friction, and therefore no energy transformed to heat—then it would be impossible to trap the ball in the valley, since a ball that initially made it over the first hill would make it over that hill again on the way back. But no real device is actually 100% efficient, a fact of nature expressed in the Second Law of Thermodynamics.

Try different kinds of balls on the track, such as a different-sized marbles, a ping-pong ball, or a golf ball.

Can you devise a relatively simple way to determine a value for the percent efficiency of the device?