Physics 105 - How Things Work - Fall, 2000

Problem Set #4 – Mechanics and Fluids

After twenty years in television’s attic, it’s back: Roller Derby! In this highly intellectual sport, two teams of gentle, friendly people roller-skate furiously around a loop track, pounding each other into the railings and the floor in an effort to… to… well, I’m not sure what they’re trying to do other than to justify some serious mayhem on wheels.

1. For simplicity, let’s suppose that the skating track is exactly circular and that it is also flat and horizontal—no banked curves. In this situation, the skaters zoom around in a circle with only friction acting to help them turn. If the track and the roller skates both have carefully chosen surfaces, the static frictional force that a skater can obtain from the skates is about equal in amount to the skater’s weight. (a) What is the fastest a skater could accelerate sideways on such a track, and (b) what effect (if any) would this acceleration limit have on the skater’s maximum speed around the track?

2. While a skater is circling around this flat, horizontal track, she is tripped up by an opponent and loses contact with the floor. (a) Which way will she travel while she is in the air? (Select a letter from the drawing below.) (b) While she is in the air, which way will she be accelerating? (Don't feel compelled to use the letters for this part).

3. In reality, the track is not level—it has sloping, banked curves. The outer edge of each banked curve is higher than the inner edge of that curve. (a) Why does this slope decrease the need for frictional forces between the skates and track, and (b) why must the slope be increased to accommodate faster skaters?

4. Suppose the track designers have just changed the track and its surface in such a way that each skater now experiences twice as much centripetal force as before. How much faster or slower would the skaters have to travel in order to follow a circular path around this modified but still circular track?

Rather than taking your annual family reunion cruise on a conventional cruise ship, your cousin has booked the cruise on a surplus Russian submarine. Fortunately, none of your relatives suffer from claustrophobia.

5. The submarine's pressure hull is essentially a long, rigid tube that can withstand the pressures of deep water. Outside this hull, protected by a thin shell, are some steel tanks that can contain either water or air. While you all are boarding the submarine, these external tanks are full of air. But when it's time to dip below the surface, the captain withdraws that air into the pressure hull and allows the surrounding water to enter the tanks. The submarine begins to sink. When it's time to return to the surface, the captain lets compressed air from inside the pressure hull flow into the external tanks and this air expels the water. The ship then floats. Why does transferring air from inside the pressure hull to the tanks outside that hull cause the ship to begin floating?

6. During the trip, one of the external tanks on this decaying boat breaks loose, so the captain can no longer replace its water with air. The captain says not to worry. He says that simply releasing some of the compressed air from within the pressure hull will cause the ship to float anyway. He's right. Why does releasing air from this rigid metal bubble and letting it flow into the surrounding ocean cause the submarine to begin floating?

7. Your aunt is enjoying the view through the periscope when she spots a couple of movie stars sunning themselves on the beach of a small private island. When she announces her discovery, everyone rushes up to the top deck to take a look. As a result of this shift in weight, the submarine begins to tip dangerously to one side and the captain asks people to move quickly to the lower decks to keep the boat from turning upside-down. In terms of potential energy, (a) why did the movement cause the submarine to begin turning over, and (b) why does the problem go away once everyone returns to the lower desks?

8. Your younger brother takes his turn at the helm and begins driving the submarine at full speed through an obstacle course of reefs. Everyone is terrified but he does an excellent job and all goes well. Each time he makes a sharp turn to the left, all of you find yourselves leaning toward the left to keep from falling over. In terms of the forces on your feet, why does leaning in the direction the boat is turning help you stay on your feet?