1. Since you love orange juice, you install plumbing from the orange juice container in the kitchen refrigerator to the breakfast area nearby. The spigot that delivers this orange juice is at exactly the same height as the orange juice container. If the container and spigot were connected only by a simple pipe, no orange juice would flow when you opened the spigot. Using the concept of pressure, briefly explain (A) why the juice wouldn't flow and (B) what is required to start the orange juice flowing.

Answer: (A) With no pressure difference (and no height difference) between the two ends of the pipe, the orange juice inside will not accelerate and will not begin to flow. (B) To get the juice flowing, there must be a pressure difference between the ends of the pipe (and/or a height difference).

Why: Like any fluid, orange juice has inertia. It won't start flowing unless it experiences a net force. In this case, that net force would have to come either from a pressure difference between the two ends of the pipe or a height difference. Since there is no height difference and also no pressure difference, the orange juice remains stationary when you open the spigot.

2. You have a small pump installed in the orange juice container and now juice flows when you open the spigot. Assuming the pump uses its electric power perfectly efficiently, how is the total amount of energy consumed by the pump related to the total amount of juice that flows out of the spigot?

Answer: The total energy consumed is proportional to the total amount of juice delivered.

Why: The work that the pump does on the juice is equal to the pressure increase the juice experiences as it passes through the pump times the volume of juice leaving that pump. This observation follows from the fact that work is force times distance in the direction of that force and that delivery volume and the pressure rise in the pump are directly related to this force times distance quantity. If the amount of juice that the pump delivers doubles, then the pump does twice as much work and consumes twice as much total energy making that delivery.

3. You also plumb the orange juice over to the guest apartment in another wing of the house. The spigot there is also at the same height as the orange juice container in the main refrigerator. When you open the spigot in the guest apartment, the juice flows much more slowly than it does from the spigot in the main kitchen. The pipes and spigots in both cases are identical, except that the guest apartment is much farther from the refrigerator and the pipe leading there is necessarily much longer. Explain briefly why that longer pipe slows the flow of juice.

Answer: The juice's viscosity limits its flow through the pipe. All else being equal (same pipe diameter, pressure difference, and juice viscosity), lengthening the pipe slows the flow rate.

Why: If the juice had no viscosity, its flow rate through a pipe wouldn't depend on the length of that pipe. But orange juice does have a viscosity and this viscosity slows its flow through a pipe. As the pipe gets longer, the juice has more opportunity to rub against the walls of the pipe and waste energy. As a result, juice flows more slowly out of the distant spigot in the guest apartment than out of the nearby spigot in the main kitchen area.

4. You also plumb orange juice down to the bar area in the basement. Although this juice passes through the same pump that provides juice to the two spigots discussed above, the juice rushes out of this basement spigot at a much greater speed. It actually sprays into a glass, rather than pouring gently. This dramatic increase in the juice's kinetic energy appears to violate conservation of energy. From where does this kinetic energy come?

Answer: The juice's gravitational potential energy becomes kinetic energy as the juice descends into the basement and flows out of the spigot there.

Why: As the juice descends, its gravitational potential energy changes into pressure potential energy or kinetic energy or both, depending on the plumbing details. The key point here is that energy is not being created out of nothing. Instead, the juice is losing gravitational potential energy and gaining kinetic energy in its place.

5. On the wall beside your bed, you have installed a narrow waterfall. Italian mineral water flows from a pipe at the top of this waterfall and is collected in a basin at the bottom. The soothing sound of the falling water helps you fall asleep and whenever you are thirsty, you can simply place a cup in the falling stream and get a drink. You notice that the lower the cup is in this stream, the faster the water is moving when it enters the cup. Explain this observation in terms of the various forms of energy that water in a streamline can have.

Answer: As the water in a streamline descends, it converts its gravitational potential energy into kinetic energy and speeds up.

Why: The stream of falling water is always at atmospheric pressure, so its total energy must be evolving from mostly gravitational potential energy to mostly kinetic energy. As its kinetic energy increases, the water moves faster and you can sense this speed increase when you put the cup in the stream.

6. When water in the waterfall strikes the bottom of your cup, it essentially stops and its kinetic energy drops to nearly zero. What becomes of that energy?

Answer: The energy becomes pressure potential energy.

Why: When the water strikes the bottom of the cup and stops, it has little gravitational potential energy and almost no kinetic energy left. To conserve energy, the missing energy must take the third form: pressure potential energy. The pressure at the bottom of the cup is thus much higher than atmospheric. The downward push of this pressure on the cup itself explains why you feel the cup being shoved downward by the falling water.

7. You plumb shampoo and conditioner into the shower stall. To avoid having to install a pump, you simply put the storage containers in the attic above the shower and allow gravity to propel the fluids through the pipes. But you sensibly select very wide pipes for these two fluids. Why is a large pipe diameter so important in this case?

Answer: Both fluids have very large viscosities and would flow extremely slowly through narrow pipes.

Why: Although gravity is propelling the flow, rather than pressure difference alone, the basic observations of viscous fluid flow in pipes are still applicable. Doubling the diameter of these pipes will increase the flow rate by a factor of roughly 16.

8. When you close the shampoo tap, the pressure inside the tap is large. There is a bottle of the same shampoo on a shelf to the right of the tap and its pressure is atmospheric. Since both the shampoo in the tap and the shampoo in the bottle are motionless and at the same height, how can one be at high pressure and the other be at low pressure? Doesn't that violate Bernoulli's equation?

Answer: The two quantities of shampoo are not part of the same streamline and do not have to have the same total energies.

Why: Bernoulli's equation only applies to fluid along a single streamline in steady-state flow. It says that this fluid maintains a constant energy is it flows steadily along that streamline. But when you compare fluids in different streamlines or fluids that are not in steady-state flow, all bets are off. In the present case, the two quantities of shampoo have entirely different histories and are not part of a single streamline. As a result, they do not have to have equal energies. If one is at a higher pressure (and energy) than the other, that's just fine.