Homework Problems

1	Two cars with different masses each have the same kinetic energy. (a) If both cars have the same brakes, capable of supplying the same force, how will the stopping distances compare? Explain. (b) Compare the times required for the cars to stop.
2	In each of the following situations, is the work being done positive, negative, or zero? (a) a bull paws the ground; (b) a fishing boat pulls a net through the water behind it; (c) the water resists the motion of the net through it; (d) you stand behind a pickup truck and lower a bale of hay from the truck's bed to the ground. Explain.	[Based on a problem by Serway and Faughn.]
3	In the earth's atmosphere, the molecules are constantly moving around. Because temperature is a measure of kinetic energy per molecule, the average kinetic energy of each type of molecule is the same, e.g., the average KE of the O2 molecules is the same as the average KE of the N2 molecules. (a) If the mass of an O2 molecule is eight times greater than that of a He atom, what is the ratio of their average speeds? Which way is the ratio, i.e., which is typically moving faster? (b) Use your result from part a to explain why any helium occurring naturally in the atmosphere has long since escaped into outer space, never to return. (Helium is obtained commercially by extracting it from rocks.) You may want to do problem 20 first, for insight.
4	Weiping lifts a rock with a weight of 1.0 N through a height of 1.0 m, and then lowers it back down to the starting point. Bubba pushes a table 1.0 m across the floor at constant speed, requiring a force of 1.0 N, and then pushes it back to where it started. (a) Compare the total work done by Weiping and Bubba. (b) Check that your answers to part a make sense, using the definition of work: work is the transfer of energy. In your answer, you'll need to discuss what specific type of energy is involved in each case.
5	In one of his more flamboyant moments, Galileo wrote Who does not know that a horse falling from a height of three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury? Equally harmless would be the fall of a grasshopper from a tower or the fall of an ant from the distance of the moon. Find the speed of an ant that falls to earth from the distance of the moon at the moment when it is about to enter the atmosphere. Assume it is released from a point that is not actually near the moon, so the moon's gravity is negligible.	√
6	(a) The crew of an 18th century warship is raising the anchor. The anchor has a mass of 5000 kg. The water is 30 m deep. The chain to which the anchor is attached has a mass per unit length of 150 kg/m. Before they start raising the anchor, what is the total weight of the anchor plus the portion of the chain hanging out of the ship? (Assume that the buoyancy of the anchor and is negligible.) (b) After they have raised the anchor by 1 m, what is the weight they are raising? (c) Define y = 0 when the anchor is resting on the bottom, and y = +30 m when it has been raised up to the ship. Draw a graph of the force the crew has to exert to raise the anchor and chain, as a function of y. (Assume that they are raising it slowly, so water resistance is negligible.) It will not be a constant! Now find the area under the graph, and determine the work done by the crew in raising the anchor, in joules. (d) Convert your answer from (c) into units of kcal.	√
7	A cylinder from the 1965 Rambler's engine. The piston is shown in its pushed out position. The two bulges at the top are for the valves that let fresh air-gas mixture in. Based on a figure from Motor Service's Automotive Encyclopedia, Toboldt and Purvis. In the power stroke of a car's gasoline engine, the fuel-air mixture is ignited by the spark plug, explodes, and pushes the piston out. The exploding mixture's force on the piston head is greatest at the beginning of the explosion, and decreases as the mixture expands. It can be approximated by F = a/x, where x is the distance from the cylinder to the piston head, and a is a constant with units of N.m. (Actually a/x1.4 would be more accurate, but the problem works out more nicely with a/x!) The piston begins its stroke at x = x1, and ends at x = x2. The 1965 Rambler had six cylinders, each with a = 220 N·m, x1 = 1.2 cm, and x2 = 10.2 cm. (a) Draw a neat, accurate graph of F vs x, on graph paper. (b) From the area under the curve, derive the amount of work done in one stroke by one cylinder. (c) Assume the engine is running at 4800 r.p.m., so that during one minute, each of the six cylinders performs 2400 power strokes. (Power strokes only happen every other revolution.) Find the engine's power, in units of horsepower (1 hp=746 W). (d) The compression ratio of an engine is defined as x2/x1. Explain in words why the car's power would be exactly the same if x1 and x2 were, say, halved or tripled, maintaining the same compression ratio of 8.5. Explain why this would not quite be true with the more realistic force equation F = a/x1.4.	√
8	The magnitude of the force between two magnets separated by a distance r can be approximated as kr-3 for large values of r. The constant k depends on the strengths of the magnets and the relative orientations of their north and south poles. Two magnets are released on a slippery surface at an initial distance ri, and begin sliding towards each other. What will be the total kinetic energy of the two magnets when they reach a final distance rf ? (Ignore friction.)	∫
9	A car starts from rest at t = 0, and starts speeding up with constant acceleration. (a) Find the car's kinetic energy in terms of its mass, m, acceleration, a, and the time, t. (b) Your answer in the previous part also equals the amount of work, W, done from t = 0 until time t. Take the derivative of the previous expression to find the power expended by the car at time t. (c) Suppose two cars with the same mass both start from rest at the same time, but one has twice as much acceleration as the other. At any moment, how many times more power is being dissipated by the more quickly accelerating car? (The answer is not 2.)
10	A space probe of mass m is dropped into a previously unexplored spherical cloud of gas and dust, and accelerates toward the center of the cloud under the influence of the cloud's gravity. Measurements of its velocity allow its potential energy, U, to be determined as a function of the distance r from the cloud's center. The mass in the cloud is distributed in a spherically symmetric way, so its density, ρ(r), depends only on r and not on the angular coordinates. Show that by finding U(r), one can infer ρ(r) as follows:	∫ *
11	A rail gun is a device like a train on a track, with the train propelled by a powerful electrical pulse. Very high speeds have been demonstrated in test models, and rail guns have been proposed as an alternative to rockets for sending into outer space any object that would be strong enough to survive the extreme accelerations. Suppose that the rail gun capsule is launched straight up, and that the force of air friction acting on it is given by F = be-cx, where x is the altitude, b and c are constants, and e is the base of natural logarithms. The exponential decay occurs because the atmosphere gets thinner with increasing altitude. (In reality, the force would probably drop off even faster than an exponential, because the capsule would be slowing down somewhat.) Find the amount of kinetic energy lost by the capsule due to air friction between when it is launched and when it is completely beyond the atmosphere. (Gravity is negligible, since the air friction force is much greater than the gravitational force.) R	∫
12	A certain binary star system consists of two stars with masses m1 and m2, separated by a distance b. A comet, originally nearly at rest in deep space, drops into the system and at a certain point in time arrives at the midpoint between the two stars. For that moment in time, find its velocity, v, symbolically in terms of b, m1, m2, and fundamental constants. [Numerical check: For m1 = 1.5 × 1030 kg, m2 = 3.0×1030 kg, and b = 2.0×1011 m you should find v = 7.7×104 m/s.]
13	An airplane flies in the positive direction along the x axis, through crosswinds that exert a force F = (a + bx)◯ + (c + dx)ŷ. Find the work done by the wind on the plane, and by the plane on the wind, in traveling from the origin to position x.	∫
14	In 1935, Yukawa proposed an early theory of the force that held the neutrons and protons together in the nucleus. His equation for the potential energy of two such particles, at a center-tocenter distance r, was PE(r) = gr-1e-r/a, where g parametrizes the strength of the interaction, e is the base of natural logarithms, and a is about 10-15 m. Find the force between two nucleons that would be consistent with this equation for the potential energy.	∫
15	Prove that the dot product defined in section 3.7 is rotationally invariant in the sense of book 1, section 7.5.
16	Fill in the details of the proof of A·B = AxBx+AyBy+AzBz on page 65.
17	Does it make sense to say that work is conserved?	Solution, p. 160
18	(a) Suppose work is done in one-dimensional motion. What happens to the work if you reverse the direction of the positive coordinate axis? Base your answer directly on the definition of work. (b) Now answer the question based on the W = Fd rule.
19	A microwave oven works by twisting molecules one way and then the other, counterclockwise and then clockwise about their own centers, millions of times a second. If you put an ice cube or a stick of butter in a microwave, you'll observe that the oven doesn't heat the solid very quickly, although eventually melting begins in one small spot. Once a melted spot forms, it grows rapidly, while the rest of the solid remains solid. In other words, it appears based on this experiment that a microwave oven heats a liquid much more rapidly than a solid. Explain why this should happen, based on the atomic-level description of heat, solids, and liquids. (See, e.g., figure b on page 35.)
20	Starting at a distance r from a planet of mass M, how fast must an object be moving in order to have a hyperbolic orbit, i.e., one that never comes back to the planet? This velocity is called the escape velocity. Interpreting the result, does it matter in what direction the velocity is? Does it matter what mass the object has? Does the object escape because it is moving too fast for gravity to act on it? p	√

Pool balls exchange momentum