This unlikely-looking contraption is in neutral equilibrium when equal weights are placed onto the two outer arms, as shown, so it will remain at rest in any position. If a net weight is placed on either side, that side will go down.

Ask your students what they think will happen when the system is released in the configuration photographed at the left above. Draw attention to the similarity between the Roberval balance and the simple pan balance.

In the pulley system photographed, the weight hanging from the free pulley is W, and the pulleys are approximately massless. The rope will be pulled at its free end, and passes over the free pulley, under the pulley attached to the weight, and back over the fixed pulley to support the free pulley. With what force F must you pull on the free end of the rope to just barely lift weight W off the ground: W, W/2, W/3, or "other?"

Let your students guess before having one of them try to lift weight W by pulling on the end of the rope. Note that this is a "gag" demonstration! The reasons why the system stays set up as photographed are (1) the rope is pinned to the "free" pulley, and (2) the rope loop is stretched tightly between the upper and lower pulleys, so that the friction prevents the weight of the "free" pulley from falling. A video of the "action" is available below.

This result can be determined in about twenty seconds as follows: Pulling on the free end with a force F causes a tension F throughout the rope. The result is a force 2F downward and F upward on the "free" pulley, causing it to move downward.

Two springs connected in series support a weight. Strings slightly longer than the springs are connected in parallel with each spring, as photographed. The connecting wire loop between the two springs is then removed, forming two separate parallel routes, each consisting of a spring and a string in series. Comparing the final configuration with the initial configuration, will the weight be higher, lower, or at the same vertical position?

The pictures above show the system in its initial and final configurations, as well as in detail of how the springs and strings are coupled at the center.

This demonstration is an analog to paradoxical behavior in complex series/parallel arrangements for other mechanical, hydraulic, and electrical systems. Perhaps the most notable is Braess' paradox for traffic flow. In certain types of congested traffic flow situations, opening an additional new route between two points may actually increase the average time taken to travel between the two points.

Engagement Suggestion:

Background:

The ball on the straight track retains essentially the same velocity and the same kinetic energy throughout the length of its run, the kinetic energy from the spring. The ball on the dipped track, however, has a more complex path. When it goes downhill, it gains kinetic energy from gravitational potential, accelerating it. It travels along the lower section of track with this increased kinetic energy, and thus greater velocity. The ball then goes uphill again, losing that additional kinetic energy – it has returned to the same height, so the principle of conservation of energy dictates that it must return to the same gravitational potential as before, giving up kinetic energy equal to what it gained. It now has only the same kinetic energy it started with, as imparted by the spring. So its velocity is now the same as its starting velocity, and the same as the velocity of the other ball.

However, during the time it was on the lowered section track, it had greater kinetic energy and greater velocity, so it traveled that distance faster than the ball on the straight track. And thus the ball on the dipped track reaches the end first, but with the same final velocity and the same final kinetic energy.

Note: The fan spins quite fast. Don't let it hit your fingers! To connect and disconnect power, use the alligator clip wire on the rear; clip it to the fan support frame for safety when not in use.

Consider inviting students to make predictions about the cart's behaviour with and without the sail. Invite them to discuss the forces involved.

Q: How will the float move inside the water vessel as the vessel executes simple harmonic motion?

A: Surprisingly, the float will remain at rest in the water vessel as it oscillates. It will even remain at rest when the vessel is stopped suddenly with your hand! See video 2 below. As the vessel oscillates, the weight density of both the floater and the water bath vary together, as the acceleration of the vessel varies, so the ratio of their densities remains the same, and they will continue to float with the same geometrical relationship!!

Q: When you remove the clamp, what will happen?: (a) the small balloon will get smaller and the large one larger, (b) the two balloons will become equal, or (c) they will stay the way they are.

A: The small balloon will blow up the larger one, and get smaller, due to surface tension effects. The rubber is thicker in a smaller balloon, and thus produces greater surface tension.