Homework Problems

1	Roy has a mass of 60 kg. Laurie has a mass of 65 kg. They are 1.5 m apart. (a) What is the magnitude of the gravitational force of the earth on Roy? (b) What is the magnitude of Roy's gravitational force on the earth? (c) What is the magnitude of the gravitational force between Roy and Laurie? (d) What is the magnitude of the gravitational force between Laurie and the sun?	√
2	During a solar eclipse, the moon, earth and sun all lie on the same line, with the moon between the earth and sun. Define your coordinates so that the earth and moon lie at greater x values than the sun. For each force, give the correct sign as well as the magnitude. (a) What force is exerted on the moon by the sun? (b) On the moon by the earth? (c) On the earth by the sun? (d) What total force is exerted on the sun? (e) On the moon? (f) On the earth?	√
3	Suppose that on a certain day there is a crescent moon, and you can tell by the shape of the crescent that the earth, sun and moon form a triangle with a 135 °interior angle at the moon's corner. What is the magnitude of the total gravitational force of the earth and the sun on the moon?
4	How high above the Earth's surface must a rocket be in order to have 1/100 the weight it would have at the surface? Express your answer in units of the radius of the Earth.
5	he star Lalande 21185 was found in 1996 to have two planets in roughly circular orbits, with periods of 6 and 30 years. What is the ratio of the two planets' orbital radii?
6	In a Star Trek episode, the Enterprise is in a circular orbit around a planet when something happens to the engines. Spock then tells Kirk that the ship will spiral into the planet's surface unless they can fix the engines. Is this scientifically correct? Why?
7	(a) Suppose a rotating spherical body such as a planet has a radius r and a uniform density ρ, and the time required for one rotation is T. At the surface of the planet, the apparent accelera- tion of a falling object is reduced by acceleration of the ground out from under it. Derive an equation for the apparent acceleration of gravity, g, at the equator in terms of r, ρ, T, and G. (b) Applying your equation from a, by what fraction is your apparent weight reduced at the equator compared to the poles, due to the Earth's rotation? (c) Using your equation from a, derive an equation giving the value of T for which the apparent acceleration of gravity becomes zero, i.e. objects can spontaneously drift off the surface of the planet. Show that T only depends on ρ, and not on r. (d) Applying your equation from c, how long would a day have to be in order to reduce the apparent weight of objects at the equator of the Earth to zero? [Answer: 1.4 hours] (e) Observational astronomers have recently found objects they called pulsars, which emit bursts of radiation at regular intervals of less than a second. If a pulsar is to be interpreted as a rotating sphere beaming out a natural "searchlight" that sweeps past the earth with each rotation, use your equation from c to show that its density would have to be much greater than that of ordinary matter. (f) Theoretical astronomers predicted decades ago that certain stars that used up their sources of energy could collapse, forming a ball of neutrons with the fantastic density of sim1017 kg/m3. If this is what pulsars really are, use your equation from c to explain why no pulsar has ever been observed that flashes with a period of less than 1 ms or so.	√
8	You are considering going on a space voyage to Mars, in which your route would be half an ellipse, tangent to the Earth's orbit at one end and tangent to Mars' orbit at the other. Your spacecraft's engines will only be used at the beginning and end, not during the voyage. How long would the outward leg of your trip last? (Assume the orbits of Earth and Mars are circular.)	√
9	(a) If the earth was of uniform density, would your weight be increased or decreased at the bottom of a mine shaft? Explain. (b) In real life, objects weigh slightly more at the bottom of a mine shaft. What does that allow us to infer about the Earth?	*
10	Ceres, the largest asteroid in our solar system, is a spherical body with a mass 6000 times less than the earth's, and a radius which is 13 times smaller. If an astronaut who weighs 400 N on earth is visiting the surface of Ceres, what is her weight?	Solution, p. 284
11	Prove, based on Newton's laws of motion and Newton's law of gravity, that all falling objects have the same acceleration if they are dropped at the same location on the earth and if other forces such as friction are unimportant. Do not just say, "g = 9.8 m/s2 - it's constant." You are supposed to be proving that g should be the same number for all objects.	Solution, p. 284
12	The figure shows an image from the Galileo space probe taken during its August 1993 flyby of the asteroid Ida. Astronomers were surprised when Galileo detected a smaller object orbiting Ida. This smaller object, the only known satellite of an asteroid in our solar system, was christened Dactyl, after the mythical creatures who lived on Mount Ida, and who protected the infant Zeus. For scale, Ida is about the size and shape of Orange County, and Dactyl the size of a college campus. Galileo was unfortunately unable to measure the time, T, required for Dactyl to orbit Ida. If it had, astronomers would have been able to make the first accurate determination of the mass and density of an asteroid. Find an equation for the density, ρ, of Ida in terms of Ida's known volume, V , the known radius, r, of Dactyl's orbit, and the lamentably unknown variable T. (This is the same technique that was used successfully for determining the masses and densities of the planets that have moons.)	Solution, p. 284
13	If a bullet is shot straight up at a high enough velocity, it will never return to the earth. This is known as the escape velocity. We will discuss escape velocity using the concept of energy in the next book of the series, but it can also be gotten at using straightforward calculus. In this problem, you will analyze the motion of an object of mass m whose initial velocity is exactly equal to escape velocity. We assume that it is starting from the surface of a spherically symmetric planet of mass Mand radius b. The trick is to guess at the general form of the solution, and then determine the solution in more detail. Assume (as is true) that the solution is of the form r= kt p, where r is the object's distance from the center of the planet at time t, and k and p are constants. (a) Find the acceleration, and use Newton's second law and Newton's law of gravity to determine k and p. You should find that the result is independent of m. (b) What happens to the velocity as t approaches infinity? (c) Determine escape velocity from the Earth's surface.	√ ∫
14	Astronomers have recently observed stars orbiting at very high speeds around an unknown object near the center of our galaxy. For stars orbiting at distances of about 1014 m from the object, the orbital velocities are about 106 m/s. Assuming the orbits are circular, estimate the mass of the object, in units of the mass of the sun, 2 × 1030 kg. If the object was a tightly packed cluster of normal stars, it should be a very bright source of light. Since no visible light is detected coming from it, it is instead believed to be a supermassive black hole.
15	Astronomers have detected a solar system consisting of three planets orbiting the star Upsilon Andromedae. The planets have been named b, c, and d. Planet b's average distance from the star is 0.059 A.U., and planet c's average distance is 0.83 A.U., where an astronomical unit or A.U. is defined as the distance from the Earth to the sun. For technical reasons, it is possible to determine the ratios of the planets' masses, but their masses cannot presently be determined in absolute units. Planet c's mass is 3.0 times that of planet b. Compare the star's average gravitational force on planet c with its average force on planet b. [Based on a problem by Arnold Arons.]	Solution, p. 284
16	Some communications satellites are in orbits called geosynchronous: the satellite takes one day to orbit the earth from west to east, so that as the earth spins, the satellite remains above the same point on the equator. What is such a satellite's altitude above the surface of the earth?	Solution, p. 284
17	As is discussed in more detail in section 5.1 of book 2, tidal interactions with the earth are causing the moon's orbit to grow gradually larger. Laser beams bounced off of a mirror left on the moon by astronauts have allowed a measurement of the moon's rate of recession, which is about 1 cm per year. This means that the gravitational force acting between earth and moon is decreasing. By what fraction does the force decrease with each 27-day orbit? [Hint: If you try to calculate the two forces and subtract, your calculator will probably give a result of zero due to rounding. Instead, reason about the fractional amount by which the quantity 1/r2 will change. As a warm-up, you may wish to observe the percentage change in 1/r2 that results from changing r from 1 to 1.01. Based on a problem by Arnold Arons.]	Solution, p. 285
18	Suppose that we inhabited a universe in which, instead of Newton's law of gravity, we had where k is some constant with different units than G. (The force is still attractive.) However, we assume that a = F/m and the rest of Newtonian physics remains true, and we use a = F/m to define our mass scale, so that, e.g., a mass of 2 kg is one which exhibits half the acceleration when the same force is applied to it as to a 1 kg mass. (a) Is this new law of gravity consistent with Newton's third law? (b) Suppose you lived in such a universe, and you dropped two unequal masses side by side. What would happen? (c) Numerically, suppose a 1.0-kg object falls with an acceleration of 10 m/s2. What would be the acceleration of a rain drop with a mass of 0.1 g? Would you want to go out in the rain? (d) If a falling object broke into two unequal pieces while it fell, what would happen? (e) Invent a law of gravity that results in behavior that is the opposite of what you found in part b. [Based on a problem by Arnold Arons.]
19	(a) A certain vile alien gangster lives on the surface of an asteroid, where his weight is 0.20 N. He decides he needs to lose weight without reducing his consumption of princesses, so he's going to move to a different asteroid where his weight will be 0.10 N. The real estate agent's database has asteroids listed by mass, however, not by surface gravity. Assuming that all asteroids are spherical and have the same density, how should the mass of his new asteroid compare with that of his old one? (b) Jupiter's mass is 318 times the Earth's, and its gravity is about twice Earth's. Is this consistent with the results of part a? If not, how do you explain the discrepancy?	Solution, p. 285
20	Where would an object have to be located so that it would experience zero total gravitational force from the earth and moon?
21	The planet Uranus has a mass of 8.68 × 1025 kg and a radius of 2.56 × 104 km. The figure shows the relative sizes of Uranus and Earth. (a) Compute the ratio gU/gE, where gU is the strength of the gravitational field at the surface of Uranus and gE is the corresponding quantity at the surface of the Earth. (b) What is surprising about this result? How do you explain it?	√