1. The first version of the balloon floats at the end of two ropes, which are attached near the head’s ears. With that placement of the ropes, the head pivots easily and keeps tipping over. The hat makes the head top-heavy. Your teammates keep trying to put the hat above the head, but it only stays there for a few seconds before the head tips over again. Your whole team is quite demoralized. (A) Use concepts relating to equilibrium to explain briefly why the hat stays on top for a few seconds and then the head flips upside-down. (B) Use concepts relating to acceleration and potential energy to explain briefly why the balloon tends to flip upside-down.

Answer: (A) The balloon is in an unstable equilibrium--it can remain briefly with its hat on top, but the slightest tip leads to it flipping over. (B) When the hat begins to tip away from the top, the balloon's total potential energy decreases. Since an object accelerates in the direction that reduces its total potential energy as quickly as possible, the hat continues to descend and the balloon flips over.

Why: Unstable equilibria usually involve a maximum of total potential energy. With the hat on top, the balloon as as much gravitational potential energy as possible. As soon as the hat tips away from that equilibrium point, the total potential energy starts to decrease and the balloon accelerates in the direction of the tip. It turns faster and faster until it is upside-down.

2. To solve the flipping problem, your team initially adds a heavy weight to the bottom of the balloon. They hope to counterbalance the hat. Unfortunately, the weighted balloon doesn’t float at all. You propose an alternative solution: attach another rope to the bottom of the head. You’ll be able to use this extra rope to actively stabilize the head’s hat-on-top equilibrium. Your scheme works! Each time the head starts to turn over, a good downward pull on the new rope returns the hat to the top. Why does this technique work?

Answer: When you pull on the string, it exerts a torque on the balloon that causes the balloon to rotate in the direction that lifts the hat back on top.

Why: You are dynamically stabilizing the balloon's unstable equilibrium. Although the equilibrium remains unstable, your yanks on the string will automatically exert a restoring torque that opposes and even overcomes the torque that is trying to flip the balloon upside-down. If you pull hard enough and often enough, you can keep the hat on top, despite its tendency to tip the balloon over.

3. To test the canvas balloon for leaks, you fill it with ordinary air. Fortunately, it proves to be airtight. You then replace the air with helium at the same pressure and temperature. Compare the number of individual air particles the balloon contains when it is full of air to the number of individual helium atoms the balloon contains when it is full of helium.

Answer: The number of particles is the same in the two cases.

Why: As long as they have the same temperature and pressure, a liter of helium and a liter of air have the same number of particles in them. Amazingly enough, a tiny helium atom is just as good at creating pressure as a much beefier air molecule. That's because, given equal temperatures, the helium atom travels much faster than the air molecule. The helium atom hits surfaces more often and its increased speed partly makes up for its smaller of mass. Overall, the little helium atom does as good a job of creating pressure as any other gas particle.

4. The properly inflated helium balloon floats nicely in air. You realize as you watch it float that there are a number of different densities present in the situation: the density of the air, the density of the helium, the density of the canvas, and the average density of the entire balloon (both canvas and helium). Put those four densities in order, from lowest density to highest density.

Answer: Lowest density: helium, 2nd lowest density: average of entire balloon, 3rd lowest density: air, highest density: canvas.

Why: Helium clearly has the lowest density; its particles weigh so little that at normal temperature and pressure, it's light stuff. Canvas, being a solid, has a very high density compared with everything else in this story. As for air's density and the balloon's average density, now we have to recognize that the balloon is a floater. For it to float, the balloon must weigh less than the air it displaces and it must therefore have an average density that's less than the average density of the fluid it displaces. So the balloon's average density is less than the density of air.

5. The cartoon character’s trademark activity is squirting water out between its teeth. Your team decides to reproduce this effect by sending water up to the balloon through a clear tube and letting that water pour out of the balloon’s mouth. After a little construction, you are able to get a steady flow of water pouring down from the balloon—steady-state flow. You note that in this situation, the water’s total energy-per-liter doesn’t change as it travels up through the pipe, out through the mouth, and back down toward the ground. Identify the changes in form that the energy undergoes as it rises and falls in this situation, ignoring air resistance and frictional effects in the tube. Also, don't worry about what happens when the water finally hits the ground.

Answer: The energy starts as pressure potential energy at the bottom of the tube. It becomes gravitational potential as the water rises in the tube. And it becomes kinetic energy as the water falls out of the top end of the tube.

Why: Since energy is conserved and none is wasted in this situation, it must simply change forms. As the water rises in the tube, it converts its pressure potential energy into gravitational potential energy. The water reaches its maximum of gravitational potential energy and then begins to fall, thereby converting that gravitational potential energy into kinetic energy.

6. With the tube full of water, the balloon is having some trouble floating. Someone suggests doubling the amount of helium gas in the balloon--pumping in twice as many helium atoms while leaving the balloon's volume unchanged. Unfortunately, this scheme won’t make the balloon float better. (A) Why won't it help the balloon float? (B) Why might it cause the balloon to pop?

Answer: (A) The added helium atoms will simple increase the balloon's weight. (B) The added helium atoms will raise the pressure inside the balloon.

Why: More of a good thing is not always better. Helium is not "antigravity", it's just very light gas. Helium allows a balloon to float by supporting it against the crushing effects of atmospheric pressure without giving it much weight. Adding more helium to the balloon than it needs to support the atmosphere is worse than a waste of helium. It actually makes the balloon heavier and it will tend to pop the balloon as well.

7. Another person suggests replacing all the helium gas in the balloon with hydrogen gas. An individual hydrogen molecule weighs approximately half as much as an individual helium atom. This person claims incorrectly that hydrogen gas can lift twice as much weight as helium gas. Explain briefly why a certain volume of hydrogen gas does not have twice the lifting capacity of an equal volume of helium gas.

Answer: The lifting capacity of a balloon is equal to the buoyant force upward on it minus its own weight. By replacing its helium with hydrogen, you can reduce its weight slightly, but the difference between the buoyant force and its weight will probably not double as a result.

Why: The balloon's lifting capacity is a subtraction problem, not a multiplication or division problem. The balloon's ability to lift a payload depends on the difference between the upward buoyant force it experiences and its own downward weight. Its weigh has already been reduced dramatically by replacing air with helium because helium weighs just 16% of air's weight. Replacing that helium with hydrogen, which weighs just 8% of air's weight, will produce only a tiny change in comparison. Both gases are nearly weightless, so the exact choice almost doesn't matter.

8. You solve the floating problem by using a narrower tube for the water, so that it doesn’t weigh as much. However, this narrower tube doesn’t carry as much water per second as the wider tube did. To compensate, you raise the total energy of the water in the tube. That scheme works and your float is a great success. Give a simple explanation for why this rise in total energy per liter makes more water flow through the narrow tube.

Answer: The amount of water that leaves the open end of the tube each second is proportional to the speed of that water. To get more water to flow out each second, you can increase the speed and that means increasing the water's kinetic energy. Since energy is conserved, increasing the water's total energy will lead to an increase in the water's kinetic energy, speed, and flow at the tube's opening.

Why: If the water travels 1 foot-per-second as it flows out of the tube's open end, then each second the tube emits a 1-foot-long cylinder of water that has a diameter equal to the tube's opening. If you want more water to flow out of this same tube each second, you'll have to speed it up. Then it can emit a 2-foot-long cylinder each second, for example. That increased speed requires increased kinetic energy. Because of energy conservation and the nature of flowing fluids, all you have to do is increase the water's total energy upstream and it will become increased kinetic energy by the time the water flows out of the tube.