Solved: Find an expression for the kinetic energy of a

System A has masses m and m separated by a distance r; system B has masses m and 2m separated by a distance 2r; system C has masses 2m and 3m separated by a distance 2r; and system D has masses 4m and 5m separated by a distance 3r. Rank these systems in order of increasing gravitational force. Indicate ties where appropriate.

In each hand you hold a 0.16-kg apple. What is the gravitational force exerted by each apple on the other when their separation is (a) 0.25 m and (b) 0.50 m?

A6.1-kg bowling ball and a 7.2-kg bowling ball rest on a rack 0.75 m apart. (a) What is the force of gravity exerted on each of the balls by the other ball? (b)At what separation is the force of gravity between the balls equal to ?

A communications satellite with a mass of 480 kg is in a circular orbit about the Earth. The radius of the orbit is 35,000 km as measured from the center of the Earth. Calculate (a) the weight of the satellite on the surface of the Earth and (b) the gravitational force exerted on the satellite by the Earth when it is in orbit.

The Attraction of Ceres Ceres, the largest asteroid known, has a mass of roughly . If Ceres passes within 14,000 km of the spaceship in which you are traveling, what force does it exert on you? (Use an approximate value for your mass, and treat yourself and the asteroid as point objects.)

In one hand you hold a 0.11-kg apple, in the other hand a 0.24kg orange. The apple and orange are separated by 0.85 m. What is the magnitude of the force of gravity that (a)the orange exerts on the apple and (b) the apple exerts on the orange?

A spaceship of mass m travels from the Earth to the Moon along a line that passes through the center of the Earth and the center of the Moon. (a)At what distance from the center of the Earth is the force due to the Earth twice the magnitude of the force due to the Moon? (b) How does your answer to part (a) depend on the mass of the spaceship? Explain.

At new moon, the Earth, Moon, and Sun are in a line, as indicated in Figure 1221. Find the direction and magnitude of the net gravitational force exerted on (a) the Earth, (b) the Moon, and (c) the Sun.

When the Earth, Moon, and Sun form a right triangle, with the Moon located at the right angle, as shown in Figure 1222, the Moon is in its third-quarter phase. (The Earth is viewed here from above its North Pole.) Find the magnitude and direction of the net force exerted on the Moon. Give the direction relative to the line connecting the Moon and the Sun.

Repeat the previous problem, this time nding the magnitude and direction of the net force acting on the Sun. Give the direction relative to the line connecting the Sun and the Moon.

Three 6.75-kg masses are at the corners of an equilateral triangle and located in space far from any other masses. (a) If the sides of the triangle are 1.25 m long, nd the magnitude of the net force exerted on each of the three masses. (b) How does your answer to part (a) change if the sides of the triangle are doubled in length?

Four masses are positioned at the corners of a rectangle, as indicated in Figure 1223. (a) Find the magnitude and direction of the net force acting on the 2.0-kg mass. (b)How do your answers to part (a) change (if at all) if all sides of the rectangle are doubled in length?

Supposed that three astronomical objects (1, 2, and 3) are observed to lie on a line, and that the distance from object 1 to object 3 is D. Given that object 1 has four times the mass of object 3 and seven times the mass of object 2, nd the distance between objects 1 and 2 for which the net force on object 2 is zero

Find the acceleration due to gravity on the surface of (a)Mercury and (b) Venus.

At what altitude above the Earths surface is the acceleration due to gravity equal to g/2?

Two 6.7-kg bowling balls, each with a radius of 0.11 m, are in contact with one another. What is the gravitational attraction between the bowling balls?

What is the acceleration due to Earths gravity at a distance from the center of the Earth equal to the orbital radius of the Moon?

Gravity on TitanTitan is the largest moon of Saturn and the only moon in the solar system known to have a substantial atmosphere. Find the acceleration due to gravity on Titans surface, given that its mass is and its radius is 2570 km.

At a certain distance from the center of the Earth, a 4.6-kg object has a weight of 2.2 N. (a) Find this distance. (b) If the object is released at this location and allowed to fall toward the Earth, what is its initial acceleration? (c) If the object is now moved twice as far from the Earth, by what factor does its weight change? Explain. (d) By what factor does its initial acceleration change? Explain.

The acceleration due to gravity on the Moons surface is known to be about one-sixth the acceleration due to gravity on the Earth. Given that the radius of the Moon is roughly onequarter that of the Earth, nd the mass of the Moon in terms of the mass of the Earth.

An Extraterrestrial Volcano Several volcanoes have been observed erupting on the surface of Jupiters closest Galilean moon, Io. Suppose that material ejected from one of these volcanoes reaches a height of 5.00 km after being projected straight upward with an initial speed of 134 m/s. Given that the radius of Io is 1820 km, (a)outline a strategy that allows you to calculate the mass of Io. (b) Use your strategy to calculate Ios mass.

Vernes Trip to the Moon In his novel From the Earth to the Moon, Jules Verne imagined that astronauts inside a spaceship would walk on the oor of the cabin when the force exerted on the ship by the Earth was greater than the force exerted by the Moon. When the force exerted by the Moon was greater, he thought the astronauts would walk on the ceiling of the cabin. (a) At what distance from the center of the Earth would the forces exerted on the spaceship by the Earth and the Moon be equal? (b) Explain why Vernes description of gravitational effects is incorrect.

Consider an asteroid with a radius of 19 km and a mass of . Assume the asteroid is roughly spherical. (a) What is the acceleration due to gravity on the surface of the asteroid? (b) Suppose the asteroid spins about an axis through its center, like the Earth, with a rotational period . What is the smallest value can have before loose rocks on the asteroids equator begin to y off the surface?

Predict/Explain The Speed of the Earth The orbital speed of the Earth is greatest around January 4 and least around July 4. (a) Is the distance from the Earth to the Sun on January 4 greater than, less than, or equal to its distance from the Sun on July 4? (b) Choose the best explanation from among the following: I. The Earths orbit is circular, with equal distance from the Sun at all times. II. The Earth sweeps out equal area in equal time, thus it must be closer to the Sun when it is moving faster. III. The greater the speed of the Earth, the greater its distance from the Sun.

Asatellite orbits the Earth in a circular orbit of radius r. At some point its rocket engine is red in such a way that its speed increases rapidly by a small amount. As a result, do the (a)apogee distance and (b)perigee distance increase, decrease, or stay the same?

Repeat the previous problem, only this time with the rocket engine of the satellite red in such a way as to slow the satellite.

Predict/ExplainThe EarthMoon Distance Is Increasing Laser reectors left on the surface of the Moon by the Apollo astronauts show that the average distance from the Earth to the Moon is increasing at the rate of 3.8 cm per year. (a)As a result, will the length of the month increase, decrease, or remain the same? (b) Choose the best explanation from among the following: I. The greater the radius of an orbit, the greater the period, which implies a longer month. II. The length of the month will remain the same due to conservation of angular momentum. III. The speed of the Moon is greater with increasing radius; therefore, the length of the month will be less.

Apollo Missions On Apollo missions to the Moon, the command module orbited at an altitude of 110 km above the lunar surface. How long did it take for the command module to complete one orbit?

Find the orbital speed of a satellite in a geosynchronous circular orbit above the surface of the Earth.

An Extrasolar Planet In July of 1999 a planet was reported to be orbiting the Sun-like star Iota Horologii with a period of 320 days. Find the radius of the planets orbit, assuming that Iota Horologii has the same mass as the Sun. (This planet is presumably similar to Jupiter, but it may have large, rocky moons that enjoy a relatively pleasant climate.)

Phobos, one of the moons of Mars, orbits at a distance of 9378 km from the center of the red planet. What is the orbital period of Phobos?

The largest moon in the solar system is Ganymede, a moon of Jupiter. Ganymede orbits at a distance of from the center of Jupiter with an orbital period of about . Using this information, nd the mass of Jupiter.

An Asteroid with Its Own Moon The asteroid 243 Ida has its own small moon, Dactyl. (See the photo on p. 390) (a)Outline a strategy to nd the mass of 243 Ida, given that the orbital radius of Dactyl is 89 km and its period is 19 hr. (b) Use your strategy to calculate the mass of 243 Ida.

GPS Satellites GPS (Global Positioning System) satellites orbit at an altitude of . Find (a) the orbital period, and (b) the orbital speed of such a satellite.

Two satellites orbit the Earth, with satellite 1 at a greater altitude than satellite 2. (a) Which satellite has the greater orbital speed? Explain. (b)Calculate the orbital speed of a satellite that orbits at an altitude of one Earth radius above the surface of the Earth. (c)Calculate the orbital speed of a satellite that orbits at an altitude of two Earth radii above the surface of the Earth.

Calculate the orbital periods of satellites that orbit (a)one Earth radius above the surface of the Earth and (b) two Earth radii above the surface of the Earth. (c)How do your answers to parts (a) and (b) depend on the mass of the satellites? Explain. (d) How do your answers to parts (a) and (b) depend on the mass of the Earth? Explain.

The Martian moon Deimos has an orbital period that is greater than the other Martian moon, Phobos. Both moons have approximately circular orbits. (a) Is Deimos closer to or farther from Mars than Phobos? Explain. (b) Calculate the distance from the center of Mars to Deimos given that its orbital period is .

Binary Stars Centauri A and Centauri B are binary stars with a separation of and an orbital period of . Assuming the two stars are equally massive (which is approximately the case), determine their mass.

Find the speed of Centauri Aand Centauri B, using the information given in the previous problem.

Sputnik The rst articial satellite to orbit the Earth was Sputnik I, launched October 4, 1957. The mass of Sputnik I was 83.5 kg, and its distances from the center of the Earth at apogee and perigee were 7330 km and 6610 km, respectively. Find the difference in gravitational potential energy for Sputnik I as it moved from apogee to perigee.

Predict/Explain (a) Is the amount of energy required to get a spacecraft from the Earth to the Moon greater than, less than, or equal to the energy required to get the same spacecraft from the Moon to the Earth? (b) Choose the best explanation from among the following: I. The escape speed of the Moon is less than that of the Earth; therefore, less energy is required to leave the Moon. II. The situation is symmetric, and hence the same amount of energy is required to travel in either direction. III. It takes more energy to go from the Moon to the Earth because the Moon is orbiting the Earth.

Consider the four masses shown in Figure 1223. (a)Find the total gravitational potential energy of this system. (b) How does your answer to part (a) change if all the masses in the system are doubled? (c) How does your answer to part (a) change if, instead, all the sides of the rectangle are halved in length?

Calculate the gravitational potential energy of a 8.8-kg mass (a) on the surface of the Earth and (b) at an altitude of 350 km. (c) Take the difference between the results for parts (b) and (a), and compare with mgh, where .

Two 0.59-kg basketballs, each with a radius of 12 cm, are just touching. How much energy is required to change the separation between the centers of the basketballs to (a)1.0 m and (b)10.0 m? (Ignore any other gravitational interactions.)

Find the minimum kinetic energy needed for a 39,000-kg rocket to escape (a) the Moon or (b) the Earth.

Predict/Explain Suppose the Earth were to suddenly shrink to half its current diameter, with its mass remaining constant. (a) Would the escape speed of the Earth increase, decrease, or stay the same? (b) Choose the best explanation from among the following: I. Since the radius of the Earth would be smaller, the escape speed would also be smaller. II. The Earth would have the same amount of mass, and hence its escape speed would be unchanged. III. The force of gravity would be much stronger on the surface of the compressed Earth, leading to a greater escape speed.

Is the energy required to launch a rocket vertically to a height h greater than, less than, or equal to the energy required to put the same rocket into orbit at the height h? Explain.

Suppose one of the Global Positioning System satellites has a speed of 4.46 km/s at perigee and a speed of 3.64 km/s at apogee. If the distance from the center of the Earth to the satellite at perigee is , what is the corresponding distance at apogee?

Meteorites from Mars Several meteorites found in Antarctica are believed to have come from Mars, including the famous ALH84001 meteorite that some believe contains fossils of ancient life on Mars. Meteorites from Mars are thought to get to Earth by being blasted off the Martian surface when a large object (such as an asteroid or a comet) crashes into the planet. What speed must a rock have to escape Mars?

Referring to Example 121, if the Millennium Eagleis at rest at point A, what is its speed at point B?

What is the launch speed of a projectile that rises vertically above the Earth to an altitude equal to one Earth radius before coming to rest momentarily?

Aprojectile launched vertically from the surface of the Moon rises to an altitude of 365 km. What was the projectiles initial speed?

Find the escape velocity for (a) Mercury and (b) Venus.

Halleys Comet Halleys comet, which passes around the Sun every 76 years, has an elliptical orbit. When closest to the Sun (perihelion) it is at a distance of and moves with a speed of 54.6 km/s. The greatest distance between Halleys comet and the Sun (aphelion) is (a) Is the speed of Halleys comet greater than or less than 54.6 km/s when it is at aphelion? Explain. (b) Calculate its speed at aphelion.

The End of the Lunar ModuleOn Apollo Moon missions, the lunar module would blast off from the Moons surface and dock with the command module in lunar orbit. After docking, the lunar module would be jettisoned and allowed to crash back onto the lunar surface. Seismometers placed on the Moons surface by the astronauts would then pick up the resulting seismic waves. Find the impact speed of the lunar module, given that it is jettisoned from an orbit 110 km above the lunar surface moving with a speed of 1630 m/s.

If a projectile is launched vertically from the Earth with a speed equal to the escape speed, how high above the Earths surface is it when its speed is half the escape speed?

Suppose a planet is discovered orbiting a distant star. If the mass of the planet is 10 times the mass of the Earth, and its radius is one-tenth the Earths radius, how does the escape speed of this planet compare with that of the Earth?

A projectile is launched vertically from the surface of the Moon with an initial speed of 1050 m/s. At what altitude is the projectiles speed one-half its initial value?

To what radius would the Sun have to be contracted for its escape speed to equal the speed of light? (Black holes have escape speeds greater than the speed of light; hence we see no light from them.)

Two baseballs, each with a mass of 0.148 kg, are separated by a distance of 395 m in outer space, far from any other objects. (a) If the balls are released from rest, what speed do they have when their separation has decreased to 145 m? (b) Suppose the mass of the balls is doubled. Would the speed found in part (a) increase, decrease, or stay the same? Explain.

On Earth, a person can jump vertically and rise to a height h. What is the radius of the largest spherical asteroid from which this person could escape by jumping straight upward? Assume that each cubic meter of the asteroid has a mass of 3500 kg.

As will be shown in Problem 63, the magnitude of the tidal force exerted on an object of mass m and length a is approximately . In this expression, M is the mass of the body causing the tidal force and r is the distance from the center of m to the center of M. Suppose you are 1 million miles away from a black hole whose mass is a million times that of the Sun. (a) Estimate the tidal force exerted on your body by the black hole. (b) At what distance will the tidal force be approximately 10 times greater than your weight?

Adumbbell has a mass m on either end of a rod of length 2a. The center of the dumbbell is a distance r from the center of the Earth, and the dumbbell is aligned radially. If , show that the difference in the gravitational force exerted on the two masses by the Earth is approximately . (Note: The difference in force causes a tension in the rod connecting the masses. We refer to this as a tidal force.) [Hint: Use the fact that ~ for .]

Referring to the previous problem, suppose the rod connecting the two masses mis removed. In this case, the only force between the two masses is their mutual gravitational attraction. In addition, suppose the masses are spheres of radius aand mass that touch each other. (The Greek letter stands for the density of the masses.) (a) Write an expression for the gravitational force between the masses m.(b)Find the distance from the center of the Earth, r, for which the gravitational force found in part (a) is equal to the tidal force found in Problem 63. This distance is known as the Roche limit. (c) Calculate the Roche limit for Saturn, assuming (The famous rings of Saturn are within the Roche limit for that planet. Thus, the innumerable small objects, composed mostly of ice, that make up the rings will never coalesce to form a moon.)

You weigh yourself on a scale inside an airplane ying due east above the equator. If the airplane now turns around and heads due west with the same speed, will the reading on the scale increase, decrease, or stay the same? Explain.

Rank objects A, B, and C in Figure 1224 in order of increasing net gravitational force experienced by the object. Indicate ties where appropriate.

Referring to Figure 1224, rank objects A, B, and C in order of increasing initial acceleration each would experience if it alone were allowed to move. Indicate ties where appropriate.

When the Moon is in its new-moon position (directly between the Earth and the Sun), does the net force exerted on it by the Sun and the Earth point toward the Sun, or point toward the Earth? Explain. (Refer to Conceptual Questions 12 and 13 as well as Figure 1220.)

A satellite goes through one complete orbit of the Earth. (a) Is the net work done on it by the Earths gravitational force positive, negative, or zero? Explain. (b) Does your answer to part (a) depend on whether the orbit is circular or elliptical?

The Crash of SkylabSkylab, the largest spacecraft ever to fall back to the Earth, met its ery end on July 11, 1979, after ying directly over Everett, WA, on its last orbit. On the CBS Evening News the night before the crash, anchorman Walter Cronkite, in his rich baritone voice, made the following statement: NASA says there is a little chance that Skylab will land in a populated area. After the commercial, he immediately corrected himself by saying,I meant to say there is little chance Skylab will hit a populated area. In fact, it landed primarily in the Indian Ocean off the west coast of Australia, though several pieces were recovered near the town of Esperance, Australia, which later sent the U.S. State Department a $400 bill for littering. The cause of Skylabs crash was the friction it experienced in the upper reaches of the Earths atmosphere. As the radius of Skylabs orbit decreased, did its speed increase, decrease, or stay the same? Explain.

Consider a system consisting of three masses on the x axis. Mass is at ; mass is at ; and mass is at . What is the total gravitational potential energy of this system?

An astronaut exploring a distant solar system lands on an unnamed planet with a radius of 3860 km. When the astronaut jumps upward with an initial speed of 3.10 m/s, she rises to a height of 0.580 m. What is the mass of the planet?

When the Moon is in its third-quarter phase, the Earth, Moon, and Sun form a right triangle, as shown in Figure 1222. Calculate the magnitude of the force exerted on the Moon by (a)the Earth and (b)the Sun. (c)Does it make more sense to think of the Moon as orbiting the Sun, with a small effect due to the Earth, or as orbiting the Earth, with a small effect due to the Sun?

An equilateral triangle 10.0 m on a side has a 1.00-kg mass at one corner, a 2.00-kg mass at another corner, and a 3.00-kg mass at the third corner (Figure 1225). Find the magnitude and direction of the net force acting on the 1.00-kg mass.

Suppose that each of the three masses in Figure 1225 is replaced by a mass of 5.95 kg and radius 0.0714 m. If the balls are released from rest, what speed will they have when they collide at the center of the triangle? Ignore gravitational effects from any other objects.

A Near Miss! In the early morning hours of June 14, 2002, the Earth had a remarkably close encounter with an asteroid the size of a small city. The previously unknown asteroid, now designated 2002 MN, remained undetected until three days after it had passed the Earth. At its closest approach, the asteroid was 73,600 miles from the center of the Earthabout a third of the distance to the Moon. (a)Find the speed of the asteroid at closest approach, assuming its speed at innite distance to be zero and considering only its interaction with the Earth. (b) Observations indicate the asteroid to have a diameter of about 2.0 km. Estimate the kinetic energy of the asteroid at closest approach, assuming it has an average density of . (For comparison, a 1-megaton nuclear weapon releases about of energy.)

Suppose a planet is discovered that has the same amount of mass in a given volume as the Earth, but has half its radius. (a) Is the acceleration due to gravity on this planet more than, less than, or the same as the acceleration due to gravity on the Earth? Explain. (b) Calculate the acceleration due to gravity on this planet.

Suppose a planet is discovered that has the same total mass as the Earth, but half its radius. (a) Is the acceleration due to gravity on this planet more than, less than, or the same as the acceleration due to gravity on the Earth? Explain. (b) Calculate the acceleration due to gravity on this planet.

Show that the speed of a satellite in a circular orbit a height h above the surface of the Earth is

In a binary star system, two stars orbit about their common center of mass, as shown in Figure 1226. If , what is the ratio of the masses m2/m1 of the two stars?

Find the orbital period of the binary star system described in the previous problem.

Using the results from Problem 54, nd the angular momentum of Halleys comet (a) at perihelion and (b) at aphelion. (Take the mass of Halleys comet to be .)

Exploring Mars In the not-too-distant future astronauts will travel to Mars to carry out scientic explorations. As part of their mission, it is likely that a geosynchronous satellite will be placed above a given point on the Martian equator to facilitate communications. At what altitude above the surface of Mars should such a satellite orbit? (Note: The Martian day is 24.6229 hours. Other relevant information can be found in Appendix C.)

A satellite is placed in Earth orbit 1000 miles higher than the altitude of a geosynchronous satellite. Referring to Active Example 121, we see that the altitude of the satellite is 23,300 mi. (a) Is the period of this satellite greater than or less than 24 hours? (b) As viewed from the surface of the Earth, does the satellite move eastward or westward? Explain. (c)Find the orbital period of this satellite.

Find the speed of the Millennium Eagleat point Ain Example 121 if its speed at point B is 0.905 m/s.

Show that the force of gravity between the Moon and the Sun is always greater than the force of gravity between the Moon and the Earth.

The astronomical unit AU is dened as the mean distance from the Sun to the Earth . Apply Keplers third law (Equation 127) to the solar system, and show that it can be written as In this expression, the period T is measured in years, the distance r is measured in astronomical units, and the constant C has a magnitude that you must determine.

(a) Find the kinetic energy of a 1720-kg satellite in a circular orbit about the Earth, given that the radius of the orbit is 12,600 miles.(b)How much energy is required to move this satellite to a circular orbit with a radius of 25,200 miles?

Space Shuttle Orbit On a typical mission, the space shuttle orbits at an altitude of 250 km above the Earths surface. (a)Does the orbital speed of the shuttle depend on its mass? Explain. (b) Find the speed of the shuttle in its orbit. (c) How long does it take for the shuttle to complete one orbit of the Earth?

Consider an object of mass m orbiting the Earth at a radius r. (a) Find the speed of the object. (b) Show that the total mechanical energy of this object is equal to times its kinetic energy. (c)Does the result of part (b) apply to an object orbiting the Sun? Explain.

In a binary star system two stars orbit about their common center of mass. Find the orbital period of such a system, given that the stars are separated by a distance dand have masses mand 2m.

Three identical stars, at the vertices of an equilateral triangle, orbit about their common center of mass (Figure 1227). Find the period of this orbital motion in terms of the orbital radius, R, and the mass of each star, M.

Find an expression for the kinetic energy of a satellite of mass m in an orbit of radius r about a planet of mass M.

Referring to Example 121, nd the xcomponent of the net force acting on the Millennium Eagleas a function of x. Plot your result, showing both negative and positive values of x.

Asatellite orbits the Earth in an elliptical orbit. At perigee its distance from the center of the Earth is 22,500 km and its speed is 4280 m/s. At apogee its distance from the center of the Earth is 24,100 km and its speed is 3990 m/s. Using this information, calculate the mass of the Earth.

Which of the two curves in Figure 1228 corresponds to comet Wild 2? A. Curve I B. Curve II

What is the mass of comet Wild 2? A. B. C. D

Find the speed needed to escape from the surface of comet Wild 2. (Note: It is easy for a person to jump upward with a speed of 3 m/s.) A. 1.6 m/s B. 2.3 m/s C. 72 m/s D. 230 m/s

Suppose comet Wild 2 had a small satellite in orbit around it, just as Dactyl orbits asteroid 243 Ida (see page 390). If this satellite were to orbit at twice the radius of the comet, what would be its period of revolution? A. 0.93 h B. 2.9 h C. 5.8 h D. 8.2 h

Find the orbital radius that corresponds to a year of 150 days.

Suppose the mass of the Sun is suddenly doubled, but the Earths orbital radius remains the same. (a) Would the length of an Earth year increase, decrease, or stay the same? (b)Find the length of a year for the case of a Sun with twice the mass. (c) Suppose the Sun retains its present mass, but the mass of the Earth is doubled instead. Would the length of the year increase, decrease, or stay the same?

Referring to Example 127(a)If the mass of the Earth were doubled, would the escape speed of a rocket increase, decrease, or stay the same? (b) Calculate the escape speed of a rocket for the case of an Earth with twice its present mass. (c) If the mass of the Earth retains its present value, but the mass of the rocket is doubled, does the escape speed increase, decrease, or stay the same?

Referring to Example 127 Suppose the Earth is suddenly shrunk to half its present radius without losing any of its mass. (a) Would the escape speed of a rocket increase, decrease, or stay the same? (b)Find the escape speed for an Earth with half its present radius.