Physics 105 - How Things Work - Fall, 2000

Problem Set #4 – Mechanics and Fluids - Solutions

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?

Answer: (a) The skater's inward acceleration can't exceed 9.8 meters/second² (the acceleration due to gravity) and (b) the skater's speed can't exceed that at which she is accelerating inward at that rate (I don't expect a specific speed value).

Why: An inward force equal to the person's weight and a downward force equal to the person's weight will both cause equal amounts of acceleration. When the force is inward, it's as though gravity were acting inward on the person, rather than downward. As a result, they "fall" inward. This inward acceleration would occur at a specific speed around the ring, namely the speed at which speed²/ring-radius = inward acceleration. If the skater tries to go faster than that speed, her inertia will prevent her from following the ring and she will crash into the outer wall of the ring.

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).

Answer: (a) She will travel along line "A" and (b) while she is in the air, she will be accelerating straight down.

Why: Her inertia will control her horizontal motion and she will coast straight ahead in the direction of her velocity before the crash. While she is in the air, she will simply be falling and will accelerate straight downward at 9.8 meters/second².

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?

Answer: (a) Because it is banked, the track's support force includes an inward (centripetal) component. That inward component helps to cause the inward (centripetal) acceleration the skater needs to follow the curving track. (b) The faster the skaters go, the more inward acceleration they experience and the more centripetal force they need. A steeper slope produces a stronger centripetal force (more horizontal component of support force).

Why: In a banked track, the track's support force points partly inward and therefore contributes to the centripetal force that accelerates skaters around the track. At higher speeds, skaters need more centripetal force so they need a steeper slope. There are also subtle issues about maintaining a constant height on the banked curve and these involve a balance between the upward component of the track's support force and skater's weight. At the right speed and right bank, the centripetal force on the skater is just enough to make her accelerate around the track and the upward support force on the skater is just enough to keep her from falling.

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?

Answer: The skaters could go about 1.414 times as fast (the square root of 2).

Why: Doubling the centripetal force will double a skater's centripetal acceleration. With twice the acceleration, the skater's speed can increase by the square root of 2. That's because the skater's centripetal acceleration during uniform circular motion is equal to the square of her speed, divided by the radius of the circle she is following.

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?

Answer: Eliminating the water in the tanks reduces the total weight of submarine. It then weighs less than the water is displaces and floats (according to Archimedes' principle).

Why: With its tanks full of water, the submarine and its contents are heavier than the water they displace. But by expelling the water with the help of compressed air, the captain manages to reduce the weight of the submarine's contents. As soon as the submarine's weight drops below that of the surrounding water, it floats upward. The buoyant force it experiences from the water exceeds its weight and it experiences a net upward force.

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?

Answer: Anything that lightens the submarine and its contents enough (while keeping their volume constant) can cause the submarine to float.

Why: As soon as the submarine's weight drops below the buoyant force exerted on it by the surrounding water, the submarine will begin to float. To lighten the submarine, they can eject anything they like: air, furniture, submariners, you name it!

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?

Answer: (a) Raising the submarine's center of gravity made it so that any tip decreases the submarine's gravitational potential energy. The boat then accelerates away from equilibrium and tips over. (b) Lowering the submarine's center of gravity makes it so that any tip increases the submarine's gravitational potential energy. The boat then accelerates toward equilibrium and remains stable.

Why: With its center of gravity too high, the boat is in an unstable equilibrium. Any disturbance causes it to begin lowering its gravitational potential energy and it accelerates in the direction that it is tipping. Over it goes. When its center of gravity is suitably low, it is in a stable equilibrium. It responds to tips by accelerating back toward its equilibrium orientation.

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?

Answer: When leaning during a turn, you make it so that the combination of support and friction forces on your feet push them directly toward your center of mass, so that you experience zero net torque and don't tip over.

Why: When you lean in the direction that you are turning, you make it so that the support torque on your feet exactly balances the friction torque on your feet. You experience zero net torque about your center of mass and don't undergo any angular acceleration. That makes it easier for you to stay upright.