FIGURE CP13.72 shows a particle of mass m at distance x

What are Keplers laws?

What is Newtons theory?

What is gravitational energy?

What does the theory say about orb

Why is the theory of gravity important?

A satellite orbits the earth with constant speed at a height above the surface equal to the earths radius. The magnitude of the satellites acceleration is a. 4gon earth b. 2gon earth c. gon earth d. 1 2 gon earth e. 1 4 gon earth f. 0

Suppose the earth suddenly came to a halt and ceased revolving around the sun. The gravitational force would then pull it directly into the sun. What would be the earths speed as it crashed?

The figure shows a binary star system. The mass of star 2 is twice the mass of star 1. Compared to F u 1 on 2, the magnitude of the force F u 2 on 1 is a. Four times as big. b. Twice as big. c. The same size. d. Half as big. e. One-quarter as big

A 1000 kg rocket is fired straight away from the surface of the earth. What speed does the rocket need to escape from the gravitational pull of the earth and never return? Assume a nonrotating earth.

A planet has four times the mass of the earth, but the acceleration due to gravity on the planets surface is the same as on the earths surface. The planets radius is a. 4Re b. 2Re c. Re d. 1 2 Re e. 1 4 Re

A less-than-successful inventor wants to launch small satellites into orbit by launching them straight up from the surface of the earth at very high speed. a. With what speed should he launch the satellite if it is to have a speed of 500 m/s at a height of 400 km? Ignore air resistance. b. By what percentage would your answer be in error if you used a flat-earth approximation?

Rank in order, from largest to smallest, the absolute values of the gravitational potential energies of these pairs of masses. The numbers give the relative masses and distances

A small supply satellite needs to dock with the International Space Station. The ISS is in a near-circular orbit at a height of 420 km. What are the speeds of the ISS and the supply satellite in this orbit?

In recent years, astronomers have discovered thousands of planets orbiting nearby stars. These are called extrasolar planets. Suppose a planet is observed to have a 1200 day period as it orbits a star at the same distance that Jupiter is from the sun. What is the mass of the star in solar masses? (1 solar mass is defined to be the mass of the sun.)

Two planets orbit a star. Planet 1 has orbital radius r1 and planet 2 has r2 = 4r1. Planet 1 orbits with period T1. Planet 2 orbits with period a. T2 = 8T1 b. T2 = 4T1 c. T2 = 2T1 d. T2 = 1 2 T1 e. T2 = 1 4 T1 f. T2 = 1 8 T

How much work must be done to boost a 1000 kg communications satellite from a low earth orbit with h = 300 km to a geosynchronous orbit?

Astronomers discover a binary star system with a period of 90 days. Both stars have a mass twice that of the sun. How far apart are the two stars?

Is the earths gravitational force on the sun larger than, smaller than, or equal to the suns gravitational force on the earth? Explain

The gravitational force of a star on orbiting planet 1 is F1. Planet 2, which is twice as massive as planet 1 and orbits at twice the distance from the star, experiences gravitational force F2. What is the ratio F1/F2?

A 1000 kg satellite and a 2000 kg satellite follow exactly the same orbit around the earth. a. What is the ratio F1/F2 of the gravitational force on the first satellite to that on the second satellite? b. What is the ratio a1/a2 of the acceleration of the first satellite to that of the second satellite?

How far away from the earth must an orbiting spacecraft be for the astronauts inside to be weightless? Explain

A space station astronaut is working outside the station as it orbits the earth. If he drops a hammer, will it fall to earth? Explain why or why not

The free-fall acceleration at the surface of planet 1 is 20 m/s2 . The radius and the mass of planet 2 are twice those of planet 1. What is g on planet 2?

Why is the gravitational potential energy of two masses negative? Note that saying because thats what the equation gives is not an explanation.

The escape speed from Planet X is 10,000 m/s. Planet Y has the same radius as Planet X but is twice as dense. What is the escape speed from Planet Y

The mass of Jupiter is 300 times the mass of the earth. Jupiter orbits the sun with TJupiter = 11.9 yr in an orbit with rJupiter = 5.2rearth. Suppose the earth could be moved to the distance of Jupiter and placed in a circular orbit around the sun. Which of the following describes the earths new period? Explain. a. 1 yr b. Between 1 yr and 11.9 yr c. 11.9 yr d. More than 11.9 yr e. It would depend on the earths speed. f. Its impossible for a planet of earths mass to orbit at the distance of Jupiter

Satellites in near-earth orbit experience a very slight drag due to the extremely thin upper atmosphere. These satellites slowly but surely spiral inward, where they finally burn up as they reach the thicker lower levels of the atmosphere. The radius decreases so slowly that you can consider the satellite to have a circular orbit at all times. As a satellite spirals inward, does it speed up, slow down, or maintain the same speed? Explain.

What is the ratio of the suns gravitational force on you to the earths gravitational force on you?

What is the ratio of the suns gravitational force on the moon to the earths gravitational force on the moon?

The centers of a 10 kg lead ball and a 100 g lead ball are separated by 10 cm. a. What gravitational force does each exert on the other? b. What is the ratio of this gravitational force to the gravitational force of the earth on the 100 g b

What is the force of attraction between a 50 kg woman and a 70 kg man sitting 1.0 m

The International Space Station orbits 300 km above the surface of the earth. What is the gravitational force on a 1.0 kg sphere inside the International Space Station

Two 65 kg astronauts leave earth in a spacecraft, sitting 1.0 m apart. How far are they from the center of the earth when the gravitational force between them is as strong as the gravitational force of the earth on one of the astronauts?

A 20 kg sphere is at the origin and a 10 kg sphere is at x = 20 cm. At what position on the x-axis could you place a small mass such that the net gravitational force on it due to the spheres is zero?

a. What is the free-fall acceleration at the surface of the sun? b. What is the free-fall acceleration toward the sun at the distance of the earth?

What is the free-fall acceleration at the surface of (a) the moon and (b) Jupiter?

A sensitive gravimeter at a mountain observatory finds that the free-fall acceleration is 0.0075 m/s2 less than that at sea level. What is the observatorys altitude?

Saturns moon Titan has a mass of 1.35 * 1023 kg and a radius of 2580 km. What is the free-fall acceleration on Titan?

A newly discovered planet has a radius twice as large as earths and a mass five times as large. What is the free-fall acceleration on its surface?

Suppose we could shrink the earth without changing its mass. At what fraction of its current radius would the free-fall acceleration at the surface be three times its present value?

Planet Z is 10,000 km in diameter. The free-fall acceleration on Planet Z is 8.0 m/s2 . a. What is the mass of Planet Z? b. What is the free-fall acceleration 10,000 km above Planet Zs north pole?

An astronaut on earth can throw a ball straight up to a height of 15 m. How high can he throw the ball on Mars?

What is the escape speed from Jupiter?

A rocket is launched straight up from the earths surface at a speed of 15,000 m/s. What is its speed when it is very far away from the earth?

A space station orbits the sun at the same distance as the earth but on the opposite side of the sun. A small probe is fired away from the station. What minimum speed does the probe need to escape the solar system?

A proposed space elevator would consist of a cable stretching from the earths surface to a satellite, orbiting far in space, that would keep the cable taut. A motorized climber could slowly carry rockets to the top, where they could be launched away from the earth using much less energy. What would be the escape speed for a craft launched from a space elevator at a height of 36,000 km? Ignore the earths rotation

Nothing can escape the event horizon of a black hole, not even light. You can think of the event horizon as being the distance from a black hole at which the escape speed is the speed of light, 3.00 * 108 m/s, making all escape impossible. What is the radius of the event horizon for a black hole with a mass 5.0 times the mass of the sun? This distance is called the Schwarzschild radiu

You have been visiting a distant planet. Your measurements have determined that the planets mass is twice that of earth but the free-fall acceleration at the surface is only one-fourth as large. a. What is the planets radius? b. To get back to earth, you need to escape the planet. What minimum speed does your rocket need?

Two meteoroids are heading for earth. Their speeds as they cross the moons orbit are 2.0 km/s. a. The first meteoroid is heading straight for earth. What is its speed of impact? b. The second misses the earth by 5000 km. What is its speed at its closest point?

A binary star system has two stars, each with the same mass as our sun, separated by 1.0 * 1012 m. A comet is very far away and essentially at rest. Slowly but surely, gravity pulls the comet toward the stars. Suppose the comet travels along a trajectory that passes through the midpoint between the two stars. What is the comets speed at the midpoint?

The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of 5.0 earth years. What are the asteroids orbital radius and speed?

You are the science officer on a visit to a distant solar system. Prior to landing on a planet you measure its diameter to be 1.8 * 107 m and its rotation period to be 22.3 hours. You have previously determined that the planet orbits 2.2 * 1011 m from its star with a period of 402 earth days. Once on the surface you find that the free-fall acceleration is 12.2 m/s2 . What is the mass of (a) the planet and (b) the star?

Three satellites orbit a planet of radius R, as shown in FIGURE EX13.26. Satellites S1 and S3 have mass m. Satellite S2 has mass 2m. Satellite S1 orbits in 250 minutes and the force on S1 is 10,000 N. a. What are the periods of S2 and S3? b. What are the forces on S2 and S3? c. What is the kinetic-energy ratio K1/K3 for S1 and S3?

A satellite orbits the sun with a period of 1.0 day. What is the radius of its orbit?

A new planet is discovered orbiting the star Vega in a circular orbit. The planet takes 55 earth years to complete one orbit around the star. Vegas mass is 2.1 times the suns mass. What is the radius of the planets orbit? Give your answer as a multiple of the radius of the earths orbit.

A small moon orbits its planet in a circular orbit at a speed of 7.5 km/s. It takes 28 hours to complete one full orbit. What is the mass of the planet?

An earth satellite moves in a circular orbit at a speed of 5500 m/s. What is its orbital period?

What are the speed and altitude of a geosynchronous satellite orbiting Mars? Mars rotates on its axis once every 24.8 hours

a. At what height above the earth is the free-fall acceleration 10% of its value at the surface? b. What is the speed of a satellite orbiting at that height?

In 2000, NASA placed a satellite in orbit around an asteroid. Consider a spherical asteroid with a mass of 1.0 * 1016 kg and a radius of 8.8 km. a. What is the speed of a satellite orbiting 5.0 km above the surface? b. What is the escape speed from the asteroid

Pluto moves in a fairly elliptical orbit around the sun. Plutos speed at its closest approach of 4.43 * 109 km is 6.12 km/s. What is Plutos speed at the most distant point in its orbit, where it is 7.30 * 109 km from the sun?

FIGURE P13.35 shows three masses. What are the magnitude and the direction of the net gravitational force on (a) the 20.0 kg mass and (b) the 5.0 kg mass? Give the direction as an angle cw or ccw from the y-axis.

What are the magnitude and direction of the net gravitational force on the 20.0 kg mass in FIGURE P13.36?

What is the total gravitational potential energy of the three masses in Figure P13.35?

What is the total gravitational potential energy of the three masses in FIGURE P13.36?

Two spherical objects have a combined mass of 150 kg. The gravitational attraction between them is 8.00 * 10-6 N when their centers are 20 cm apart. What is the mass of each?

Two 100 kg lead spheres are suspended from 100-m-long massless cables. The tops of the cables have been carefully anchored exactly 1 m apart. By how much is the distance between the centers of the spheres less than 1 m?

A rogue black hole with a mass 12 times the mass of the sun drifts into the solar system on a collision course with earth. How far is the black hole from the center of the earth when objects on the earths surface begin to lift into the air and fall up into the black hole? Give your answer as a multiple of the earths radius.

An object of mass m is dropped from height h above a planet of mass M and radius R. Find an expression for the objects speed as it hits the ground.

A projectile is shot straight up from the earths surface at a speed of 10,000 km/h. How high does it go?

The unexplored planet Alpha Centauri III has a radius of 7.0 * 106 m. A visiting astronaut drops a rock, from rest, into a 100-m-deep crevasse. She records that it takes 6.0 s for the rock to reach the bottom. What is the mass of Alpha Centauri III?

An astronaut circling the earth at an altitude of 400 km is horrified to discover that a cloud of space debris is moving in the exact same orbit as his spacecraft, but in the opposite direction. The astronaut detects the debris when it is 25 km away. How much time does he have to fire his rockets and change orbits?

Suppose that on earth you can jump straight up a distance of 45 cm. Asteroids are made of material with mass density 2800 kg/m3 . What is the maximum diameter of a spherical asteroid from which you could escape by jumping?

A rogue band of colonists on the moon declares war and prepares to use a catapult to launch large boulders at the earth. Assume that the boulders are launched from the point on the moon nearest the earth. For this problem you can ignore the rotation of the two bodies and the orbiting of the moon. a. What is the minimum speed with which a boulder must be launched to reach the earth? Hint: The minimum speed is not the escape speed. You need to analyze a three-body system. b. Ignoring air resistance, what is the impact speed on earth of a boulder launched at this minimum speed

Two spherical asteroids have the same radius R. Asteroid 1 has mass M and asteroid 2 has mass 2M. The two asteroids are released from rest with distance 10R between their centers. What is the speed of each asteroid just before they collide? Hint: You will need to use two conservation l

A starship is circling a distant planet of radius R. The astronauts find that the free-fall acceleration at their altitude is half the value at the planets surface. How far above the surface are they orbiting? Your answer will be a multiple of R

The two stars in a binary star system have masses 2.0 * 1030 kg and 6.0 * 1030 kg. They are separated by 2.0 * 1012 m. What are a. The systems rotation period, in years? b. The speed of each star?

A 4000 kg lunar lander is in orbit 50 km above the surface of the moon. It needs to move out to a 300-km-high orbit in order to link up with the mother ship that will take the astronauts home. How much work must the thrusters do

The 75,000 kg space shuttle used to fly in a 250-km-high circular orbit. It needed to reach a 610-km-high circular orbit to service the Hubble Space Telescope. How much energy was required to boost it to the new orbit?

How much energy would be required to move the earth into a circular orbit with a radius 1.0 km larger than its current radius?

NASA would like to place a satellite in orbit around the moon such that the satellite always remains in the same position over the lunar surface. What is the satellites altitud

In 2014, the European Space Agency placed a satellite in orbit around comet 67P/Churyumov-Gerasimenko and then landed a probe on the surface. The actual orbit was elliptical, but well approximate it as a 50-km-diameter circular orbit with a period of 11 days. a. What was the satellites orbital speed around the comet? (Both the comet and the satellite are orbiting the sun at a much higher speed.)b. What is the mass of the comet? c. The lander was pushed from the satellite, toward the comet, at a speed of 70 cm/s, and it then felltaking about 7 hoursto the surface. What was its landing speed? The comets shape is irregular, but on average it has a diameter of 3.6 km

A satellite orbiting the earth is directly over a point on the equator at 12:00 midnight every two days. It is not over that point at any time in between. What is the radius of the satellites orbit

FIGURE P13.57 shows two planets of mass m orbiting a star of mass M. The planets are in the same orbit, with radius r, but are always at opposite ends of a diameter. Find an exact expression for the orbital period T. Hint: Each planet feels two forces

Figure 13.17 showed a graph of log T versus log r for the planetary data given in Table 13.2. Such a graph is called a log-log graph. The scales in Figure 13.17 are logarithmic, not linear, meaning that each division along the axis corresponds to a factor of 10 increase in the value. Strictly speaking, the correct labels on the y-axis should be 7, 8, 9, and 10 because these are the logarithms of 107 ,c,1010 . a. Consider two quantities u and v that are related by the expression v p = Cuq , where C is a constant. The exponents p and q are not necessarily integers. Define x = log u and y = log v. Find an expression for y in terms of x. b. What shape will a graph of y versus x have? Explain. c. What slope will a graph of y versus x have? Explain. d. Use the experimentally determined best-fit line in Figure 13.17 to find the mass of the sun

Large stars can explode as they finish burning their nuclear fuel, causing a supernova. The explosion blows away the outer layers of the star. According to Newtons third law, the forces that push the outer layers away have reaction forces that are inwardly directed on the core of the star. These forces compress the core and can cause the core to undergo a gravitational collapse. The gravitational forces keep pulling all the matter together tighter and tighter, crushing atoms out of existence. Under these extreme conditions, a proton and an electron can be squeezed together to form a neutron. If the collapse is halted when the neutrons all come into contact with each other, the result is an object called a neutron star, an entire star consisting of solid nuclear matter. Many neutron stars rotate about their axis with a period of 1 s and, as they do so, send out a pulse of electromagnetic waves once a second. These stars were discovered in the 1960s and are called pulsars. a. Consider a neutron star with a mass equal to the sun, a radius of 10 km, and a rotation period of 1.0 s. What is the speed of a point on the equator of the star? b. What is g at the surface of this neutron star? c. A stationary 1.0 kg mass has a weight on earth of 9.8 N. What would be its weight on the star? d. How many revolutions per minute are made by a satellite orbiting 1.0 km above the surface? e. What is the radius of a geosynchronous orbit?

The solar system is 25,000 light years from the center of our Milky Way galaxy. One light year is the distance light travels in one year at a speed of 3.0 * 108 m/s. Astronomers have determined that the solar system is orbiting the center of the galaxy at a speed of 230 km/s. a. Assuming the orbit is circular, what is the period of the solar systems orbit? Give your answer in years.b. Our solar system was formed roughly 5 billion years ago. How many orbits has it completed? c. The gravitational force on the solar system is the net force due to all the matter inside our orbit. Most of that matter is concentrated near the center of the galaxy. Assume that the matter has a spherical distribution, like a giant star. What is the approximate mass of the galactic center? d. Assume that the sun is a typical star with a typical mass. If galactic matter is made up of stars, approximately how many stars are in the center of the galaxy? Astronomers have spent many years trying to determine how many stars there are in the Milky Way. The number of stars seems to be only about 10% of what you found in part d. In other words, about 90% of the mass of the galaxy appears to be in some form other than stars. This is called the dark matter of the universe. No one knows what the dark matter is. This is one of the outstanding scientific questions of our day.

Three stars, each with the mass of our sun, form an equilateral triangle with sides 1.0 * 1012 m long. (This triangle would just about fit within the orbit of Jupiter.) The triangle has to rotate, because otherwise the stars would crash together in the center. What is the period of rotation?

Comets move around the sun in very elliptical orbits. At its closet approach, in 1986, Comet Halley was 8.79 * 107 km from the sun and moving with a speed of 54.6 km/s. What was the comets speed when it crossed Neptunes orbit in 2006?

A 55,000 kg space capsule is in a 28,000-km-diameter circular orbit around the moon. A brief but intense firing of its engine in the forward direction suddenly decreases its speed by 50%. This causes the space capsule to go into an elliptical orbit. What are the space capsules (a) maximum and (b) minimum distances from the center of the moon in its new orbit? Hint: You will need to use two conservation laws

In Problems 64 through 66 you are given the equation(s) used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation(s). b. Draw a pictorial representation. c. Finish the solution of the problem.16.67 * 10-11 Nm2 /kg2 215.68 * 1026 kg2 r2 = 16.67 * 10-11 Nm2 /kg2 215.98 * 1024 kg2 16.37 * 106 m22

In Problems 64 through 66 you are given the equation(s) used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation(s). b. Draw a pictorial representation. c. Finish the solution of the problem.16.67 * 10-11 Nm2 /kg2 215.98 * 1024 kg211000 kg2 r2 = 11000 kg211997 m/s22 r

In Problems 64 through 66 you are given the equation(s) used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation(s). b. Draw a pictorial representation. c. Finish the solution of the problem.2 1100 kg2v2 2 - 16.67 * 10-11 Nm2 /kg2 217.36 * 1022 kg21100 kg2 1.74 * 106 m = 0 - 16.67 * 10-11 Nm2 /kg2 217.36 * 1022 kg21100 kg2 3.48 * 106 m

Two Jupiter-size planets are released from rest 1.0 * 1011 m apart. What are their speeds as they crash together?

A satellite in a circular orbit of radius r has period T. A satellite in a nearby orbit with radius r + r, where r V r, has the very slightly different period T + T. a. Show that T T = 3 2 r r b. Two earth satellites are in parallel orbits with radii 6700 km and 6701 km. One day they pass each other, 1 km apart, along a line radially outward from the earth. How long will it be until they are again 1 km apart?

While visiting Planet Physics, you toss a rock straight up at 11 m/s and catch it 2.5 s later. While you visit the surface, your cruise ship orbits at an altitude equal to the planets radius every 230 min. What are the (a) mass and (b) radius of Planet Physics?

A moon lander is orbiting the moon at an altitude of 1000 km. By what percentage must it decrease its speed so as to just graze the moons surface one-half period later?

Lets look in more detail at how a satellite is moved from one circular orbit to another. FIGURE CP13.71 shows two circular orbits, of radii r1 and r2, and an elliptical orbit that connects them. Points 1 and 2 are at the ends of the semimajor axis of the ellipse. a. A satellite moving along the elliptical orbit has to satisfy two conservation laws. Use these two laws to prove that the velocities at points 1 and 2 are v= 1 = B 2GM1r2/r12 r1 + r2 and v= 2 = B 2GM1r1/r22 r1 + r2 The prime indicates that these are the velocities on the elliptical orbit. Both reduce to Equation 13.22 if r1 = r2 = r. b. Consider a 1000 kg communications satellite that needs to be boosted from an orbit 300 km above the earth to a geosynchronous orbit 35,900 km above the earth. Find the velocity v1 on the inner circular orbit and the velocity v= 1 at the low point on the elliptical orbit that spans the two circular orbits. c. How much work must the rocket motor do to transfer the satellite from the circular orbit to the elliptical orbit? d. Now find the velocity v= 2 at the high point of the elliptical orbit and the velocity v2 of the outer circular orbit. e. How much work must the rocket motor do to transfer the satellite from the elliptical orbit to the outer circular orbit? f. Compute the total work done and compare your answer to the result of Example 13.6

FIGURE CP13.72 shows a particle of mass m at distance x from the center of a very thin cylinder of mass M and length L. The particle is outside the cylinder, so x 7 L/2. a. Calculate the gravitational potential energy of these two masses. b. Use what you know about the relationship between force and potential energy to find the magnitude of the gravitational force on m when it is at position x.