1. The canoe is heavy because its fiberglass skin is quite dense. A block of fiberglass normally sinks in water. In a sentence or two, explain how the canoe manages to float in water.

Answer: The canoe is hollow and is filled with air. Its average density is much less than that of water, so the buoyant force it experiences when partially submerged in water easily supports its weight.

Why: An object's ability to float on a fluid depends on that object's average density. Although the canoe has a dense skin and large total weight, most of its volume is occupied by air and its average density is small. When partially submerged in water, it experiences an upward buoyant force that balances its downward weight and it floats.

2. The stationary canoe is quite stable while you are seated properly, but becomes unstable when both of you stand up. We can understand this effect by noticing that whenever the canoe tips to the right or left, it effectively pivots about a horizontal axis that runs from the front of the canoe to the back. This axis is located roughly a foot above the bottom of the canoe. As long as the occupants are low, the canoe's total center of gravity is located below this axis and the canoe exhibits static stability. But when you both stand up and raise the center of gravity above this axis, over you go. Use the relationship between potential energy and acceleration to explain this transition from stable to unstable behavior when you stand up.

Answer: When the overall center of gravity is below the pivot axis, any tipping raises the center of gravity and increases the overall potential energy (gravitational potential energy). Since objects accelerate in the direction that reduces their total potential energy as quickly as possible, the canoe naturally accelerates back to its upright orientation. But when the center of gravity is above the pivot axis, a tip lowers the center of gravity and the overall potential energy. The top-heavy canoe thus accelerates away from upright and tips over.

Why: Static stability depends critically on how total potential energy changes with orientation or position. If that energy increases as a particular shift occurs, then the system will naturally accelerate back toward equilibrium and away from that shift. However, if the overall potential energy decreases with the shift, then the system will accelerate away from equilibrium and a catastrophic shift in that direction may easily occur.

3. Once the canoe is heading forward quickly enough, its dynamic behavior (they way it moves) begins to contribute to its overall stability. Your friend just can't seem to stay seated and keeps standing up. On one particular occasion, the top of the boat begins to tip sharply to the left. Fortunately, this tipping caused the forward-moving boat to steer in such a way that it spontaneously recovered from the tip. (A) Which way did the boat steer, toward the left or right, and (B) why did that direction of steering cause the boat to recover its upright orientation?

Answer: (A) The boat steered toward the left. (B) That steering caused the boat to drive under the boat's overall center of gravity and support it. (The boat returned to equilibrium, even though that equilibrium was an unstable one.)

Why: As long as the boat's overall center of gravity is vertically aligned with the boat's pivot axis, the boat will be in equilibrium and have no tendency to tip over. If that center of gravity is below the pivot axis, then that equilibrium is stable and even tipping the boat a little won't cause it to flip. But if that center of gravity is above the pivot axis, the equilibrium is unstable and the boat needs to maintain that equilibrium actively. That's where the "automatic" steering comes into play. By naturally steering under the center of gravity, the canoe tends to return the boat to upright, even when its equilibrium is unstable.

4. As you paddle forward, you find that the way you hold the paddle affects the feel of paddling. When paddling on the right side of the canoe, you hold the small end of the paddle stationary in your left hand and draw the large, blade end of the paddle through the water using your right hand. If your right hand is almost touching the blade, you find that you can paddle using relatively little force. However, if your right hand is more toward the middle of the paddle's handle and rather far from the blade itself, you must exert a much larger force on the paddle to pull it through the water. This would appear to violate conservation of energy, since in either case, you are doing the same task: pulling the paddle blade through the water. What other issue is present here that resolves this problem and explains why energy conservation still holds?

Answer: Your hand pulls on the paddle for a shorter distance when you grip the paddle near the middle of its handle. Overall, the work you do (force times distance) is the same in both cases.

Why: While it is true that the force you must exert on the paddle increases as you move your right hand away from the blade end of the paddle, the distance your right hand must move that part of the paddle decreases. The product of the two quantities, force times distance traveled in the direction of that force, remains the same and thus the work that you do remains the same. This trade off between force and distance is an example of mechanical advantage. You can adjust the relationship between those two quantities so that you exert a comfortable force for a comfortable distance.

5. As the day wears on, the air gets hotter and hotter. Does the canoe float at the same height in the water, or does it rise upward slightly or sink downward slightly? Why? (Note: assume that the air's pressure and constituents don't change. Also assume that the canoe's size remains exactly constant.)

Answer: The canoe rises upward slightly. That's because as the air's temperature increases, its density decreases. Therefore, the average density of the canoe and the air it contains decreases. The canoe becomes lighter and floats more easily. It displaces less water than before.

Why: The air in the boat contributes to its overall weight and its average density. All else being constant, hotter air is less dense than colder air. Therefore the canoe's average density drops as the temperature rises. With less total weight and a lower average density, the canoe is easier to float. It doesn't need to displace as much water as before.

[Note for the experts: In reality, the canoe is displacing not only water, but also air. It actually floats at a height such that the average density of the water and air that it displaces exactly equals the average density of its skin and the air it contains. Because air displacement is involved, as the air becomes less dense, the average density of the fluid that the canoe is displacing actually drops slightly. But the effect of that drop is exactly balanced by a drop in the density of the air inside the canoe between the level of the surrounding lake and the top of the canoe. Because of this cancellation above the water line, only the drop in the density of air below the water line actually matters in this problem!]

6. After crossing the lake, you and your friend put the canoe in a truck and drive it to a lake up at the top of a mountain. You put the canoe in the water there and set out across this second lake. Does the canoe float at the same height in the water, or does it rise upward slightly or sink downward slightly? Why? (Note: assume that the air's temperature and constituents are the same as they were above the first lake. Also assume that the canoe's size remains exactly constant.)

Answer: The canoe rises upward slightly because the density of the air inside it decreases and lowers the canoe's average density.

Why: Anything that lowers the canoe's average density causes it to float higher in the water because the boat then needs to displace less water to support itself. At higher altitudes, the air is supporting less weight of atmosphere overhead and its pressure is therefore less than at lower altitudes. The air above this second, high-altitude lake is lower pressure air and its density is therefore lower than the air above the first, low-altitude lake. At high altitudes, the canoe contains fewer air molecules, it weighs less, and it floats more easily.

7. While the canoe is floating on the lake, the bottom skin of the canoe has water beneath it and air above it. The water and air are separated by only a few millimeters of fiberglass. Compare the pressure of the water just beneath this bottom skin of the canoe to the pressure of air just above that skin. Are they equal or is one pressure higher than the other?

Answer: The pressure of the water beneath the skin is (significantly) higher than the pressure of the air above that skin.

Why: Although we think of it in terms of buoyancy, the canoe is actually being supported by a pressure difference between the water beneath it and the air above it. The water pushes upward harder on the bottom of the canoe's skin than the air pushes downward on the top of the canoe's skin. We can understand this pressure difference by recognizing that the water beneath the canoe's bottom is located significantly below the lake's average level. Since the water pressure at the lake's surface is atmospheric pressure and the water's pressure increases with depth, the water at the depth of the canoe's bottom skin has a pressure substantially above atmospheric pressure. In contrast, the air above the canoe's skin is essentially at what we are calling atmospheric pressure. Although this air is slightly lower in altitude than the air exactly at the lake's surface, the rise in pressure that occurs as you look deeper down in the atmosphere is much smaller than that in water. Air's density is just so small compared to that of water that air's pressure doesn't change quickly with altitude. So the air pressure above the canoe's bottom skin is essentially atmospheric pressure while the water pressure below that skin is significantly above atmospheric pressure.

8. After some splashing, your canoe has a little water inside it. You are still gliding across the lake, so you can't tip the canoe over to get rid of the water. Your friend suggests drilling a small hole in the very bottom of the canoe to let this water drain out into the lake beneath the canoe. Would water actually drain out of the canoe if you were to drill a hole in the very bottom? Why or why not?

Answer: It would not work. Water would flow into the canoe through that hole because lake water would accelerate toward the lower pressure above the canoe's bottom skin.

Why: There is a substantial difference in pressures between the water below the canoe's bottom and the air above the canoe's bottom. Since fluids accelerate toward lower pressure (we can neglect gravity here because the pressure difference overwhelms it), the lake water below the skin accelerates toward the air above the skin. The canoe would fill with water rather than emptying.