1. Devise an experiment like the one Newton performed to test the
equivalence of inertial and gravitational masses. Use different masses
on pendula of equal length to show that the period depends on the ratio
of .... Get solution
1q. Laser light from Earth is received for an experiment by an Earth satellite. Is the light redshifted or blue-shifted? What happens to the light if it is reflected back to Earth? Get solution
2. Controllers want to communicate with a satellite in orbit 480 km above Earth. If they use a signal of frequency 100 MHz, what is the gravitational redshift? Assume g is constant. Get solution
2q. Explain why true weightlessness occurs only at the center of mass of the space station as it rotates around Earth. Get solution
3. For clocks near the surface of Earth, show that Equation (15.4) reduces to Equation (15.3) for Δf /f . ... ... ... Get solution
3q. Why does a drop of water become bulged in the space station due to Earth’s gravitational field? Draw the water drop showing the direction to Earth’s center. Get solution
4. Repeat Example 15.1 using the more accurate Equation (15.4) for the gravitational redshift. Compare with the result of Example 15.1. ... ... ... ... Get solution
4q. Devise a way for the occupants of a spaceship to know whether they are being pulled into a black hole. What can they do if they determine they are within the Schwarzschild radius? Get solution
5. In Shapiro’s experiment on the time delay during the superior conjunction of Venus and Earth, how much time did it take for the radar signals to travel the round trip to Venus? What percentage change was he looking for? Get solution
5q. Astronauts riding in the space station are said to be in “zero-free” or “micro” gravity. Explain why this is not really so. Is the net force on them zero? Get solution
6. Calculate the gravitational redshift of radiation of wavelength 550 nm (the middle of the visible range) that is emitted from a neutron star having a mass of 5.8 × 1030 kg and a radius of 10 km. Assume that the radiation is being detected far from the neutron star. Get solution
6q. In the experiment discussed in Chapter 2 of the atomic clocks flown around Earth, what was the effect regarding general relativity? Get solution
7. Radiation is emitted from the sun over a wide range of wavelengths. Calculate the gravitational redshift of light of wavelength 400 nm and 700 nm (the two ends of the visible range) that is emitted from the sun and received a great distance away. Get solution
7q. In 1919 when the gravitational deflection of light was measured, why did the scientists travel to Africa and South America? Get solution
8. Assume the experiment of Pound and Rebka is performed on the top of the Empire State Building (height = 381 m). What are the change in frequency and the percentage change in frequency due to the gravitational redshift? Get solution
8q. How likely is it for a black hole to collide with Earth? Would we have much warning? Get solution
9. In the experiment of Pound and Rebka, a 14.4-keV gamma ray fell through a distance of 22.5 m near Earth’s surface. What are the change in frequency and the percentage change in frequency due to the gravitational redshift? Get solution
10. Find the relative frequency shift Δf/f for light emitted at the surface of the sun (radius 6.96 × 105 km, mass 1.99 × 1030 kg) if the light is received at (a) the planet Mercury and (b) Earth. Get solution
10q. We mention in the text that gravitational redshifts can be observed and measured during the collapse of a star into a black hole. When might the redshifts cease? Get solution
11. A He-Ne laser with wavelength 632.8 nm is fired from a great distance toward a neutron star with mass 4.5 ×1030 kg and radius 12 km. What is the wavelength of light received at the neutron star’s surface? Get solution
11q. We mentioned that astronauts can tell whether they are in outer space or “falling around Earth” by observing a drop of water in the corner of their spacecraft. What are the tidal forces that were mentioned, and where do they come from? Why are they called “tidal” on Earth? Get solution
12. What is the value of the Schwarzschild radius for the moon? (mmoon = 7.35 × 1022 kg) Get solution
12q. Why can we conclude from Equation (15.1) that the inertial and gravitational masses are equal? ... Get solution
13. Calculate the Schwarzschild radius for Jupiter. (mJupiter = 1.90 × 1027 kg) Get solution
13q. Explain why it’s expected that primordial black holes should not last for a long time before evaporating completely. Why is the last part of such a black hole’s lifetime described in the text as comparable to numerous hydrogen bomb explosions? Get solution
14. Stephen Hawking has predicted the temperature of a black hole of mass M to be ..., where k is Boltzmann’s constant. (a) Calculate the temperature of a black hole with the mass of the sun. Discuss the implications of the temperature you calculate. (b) Find the temperature of a supermassive black hole, which may exist at the center of some galaxies, with a mass 6.0 × 109 times the sun’s mass. Get solution
14q. Some concern was expressed that the high-energy particles produced by the Large Hadron Collider might generate small black holes that could grow out of control and eventually consume all of Earth’s mass. Why is this not a likely scenario? Get solution
15. Calculate the mass and Schwarzschild radius of a black hole at room temperature (see Problem 14). How many solar masses is this? Problem 14 Stephen Hawking has predicted the temperature of a black hole of mass M to be ..., where k is Boltzmann’s constant. (a) Calculate the temperature of a black hole with the mass of the sun. Discuss the implications of the temperature you calculate. (b) Find the temperature of a supermassive black hole, which may exist at the center of some galaxies, with a mass 6.0 × 109 times the sun’s mass. Get solution
16. The supermassive black hole at the center of the NGC 4261 galaxy is thought to have a mass of 1 billion suns. (a) Calculate its Schwarzschild radius and compare it with the size of our solar system. (b) How much time would this black hole take to evaporate by Hawking radiation? Get solution
17. (a) Use the known lifetime of the universe to determine the mass of a black hole that would evaporate all its mass during that time. (b) How likely is it that a black hole of this mass could exist? Get solution
18. Determine the constant α in Equation (15.10). ... Get solution
19. Because the evaporation rate of a black hole increases as the black hole’s size decreases, a small primordial black hole releases energy at a fantastic rate. Find the mass of a black hole that would release energy equivalent to a one-megaton (4.2 × 1015 J) hydrogen bomb every second. Because the evaporation rate of a black hole increases as the black hole’s size decreases, a small primordial black hole releases energy at a fantastic rate. Find the mass of a black hole that would release energy equivalent to a one-megaton (4.2 × 1015 J) hydrogen bomb every second. Get solution
20. For a black hole with the mass of our moon (7.3 × 1022 kg) find (a) its Schwarzschild radius; (b) its effective temperature; and (c) the potential energy associated with this black hole being just above Earth’s surface. Get solution
21g. The Global Positioning System satellites operate at an altitude of 20,200 km and use communication frequencies of 1575.42 MHz. Find the gravitational frequency change with respect to Earth. Get solution
22g. Derive Equation (15.4). ... Get solution
23g. One of the communication frequencies that the International Space Station uses is 259.7 MHz. (a) Find the gravitational frequency change with respect to Earth when the station is at its mean altitude of 352 km. Assume g is constant and equal to 9.80 m/s2. (b) Now do a more precise calculation of the frequency shift, without assuming that g is constant. Get solution
24g. Weightlessness occurs only at the center of mass of the International Space Station as it rotates 350 km above Earth. Calculate the effective g that an astronaut in the station who is 3 m closer to Earth than the center of mass would feel. You may choose to ignore relativistic effects. Get solution
25g. Find the mass of a particle with a Compton wavelength of πrS where rS is the Schwarzschild radius. This mass is called the Planck mass mP, and the energy required to create the mass is called the Planck energy EP = mPc 2. Determine the values of both the Planck mass and energy. Get solution
26g. The length scale on which the quantized nature of gravity should first become evident is called the Planck length. (a) Determine it using dimensional analysis using the fundamental constants G, h, and c. (b) Determine it by finding the de Broglie wavelength of the Planck mass of Problem 25. Are the values close to the value of 10-35 m? Get solution
27g. Use the fundamental constants G, h, and c and dimensional analysis to determine a time constant called the Planck time. How much time would it take light to travel the Planck length discovered in the previous problem? Are these two times consistent? Get solution
28g. A communications satellite is at a geosynchronous orbit position (35,870 km above Earth’s surface) and communicates with Earth at a frequency of 2.0 × 109 Hz. What is the frequency change due to gravity? Get solution
29g. Stephen Hawking’s derivation of the black hole temperature used the fact that the black hole’s entropy is given by ... Complete the derivation using the thermodynamic definition of temperature .... Assume that the black hole’s energy is entirely mass-energy, that is, U = Mc2. Get solution
30g. It is written in the text that light traveling horizontally across the continental United States should fall about 1 mm due to gravity. Determine the approximate vertical fall for light traveling from Los Angeles to New York City. Get solution
1q. Laser light from Earth is received for an experiment by an Earth satellite. Is the light redshifted or blue-shifted? What happens to the light if it is reflected back to Earth? Get solution
2. Controllers want to communicate with a satellite in orbit 480 km above Earth. If they use a signal of frequency 100 MHz, what is the gravitational redshift? Assume g is constant. Get solution
2q. Explain why true weightlessness occurs only at the center of mass of the space station as it rotates around Earth. Get solution
3. For clocks near the surface of Earth, show that Equation (15.4) reduces to Equation (15.3) for Δf /f . ... ... ... Get solution
3q. Why does a drop of water become bulged in the space station due to Earth’s gravitational field? Draw the water drop showing the direction to Earth’s center. Get solution
4. Repeat Example 15.1 using the more accurate Equation (15.4) for the gravitational redshift. Compare with the result of Example 15.1. ... ... ... ... Get solution
4q. Devise a way for the occupants of a spaceship to know whether they are being pulled into a black hole. What can they do if they determine they are within the Schwarzschild radius? Get solution
5. In Shapiro’s experiment on the time delay during the superior conjunction of Venus and Earth, how much time did it take for the radar signals to travel the round trip to Venus? What percentage change was he looking for? Get solution
5q. Astronauts riding in the space station are said to be in “zero-free” or “micro” gravity. Explain why this is not really so. Is the net force on them zero? Get solution
6. Calculate the gravitational redshift of radiation of wavelength 550 nm (the middle of the visible range) that is emitted from a neutron star having a mass of 5.8 × 1030 kg and a radius of 10 km. Assume that the radiation is being detected far from the neutron star. Get solution
6q. In the experiment discussed in Chapter 2 of the atomic clocks flown around Earth, what was the effect regarding general relativity? Get solution
7. Radiation is emitted from the sun over a wide range of wavelengths. Calculate the gravitational redshift of light of wavelength 400 nm and 700 nm (the two ends of the visible range) that is emitted from the sun and received a great distance away. Get solution
7q. In 1919 when the gravitational deflection of light was measured, why did the scientists travel to Africa and South America? Get solution
8. Assume the experiment of Pound and Rebka is performed on the top of the Empire State Building (height = 381 m). What are the change in frequency and the percentage change in frequency due to the gravitational redshift? Get solution
8q. How likely is it for a black hole to collide with Earth? Would we have much warning? Get solution
9. In the experiment of Pound and Rebka, a 14.4-keV gamma ray fell through a distance of 22.5 m near Earth’s surface. What are the change in frequency and the percentage change in frequency due to the gravitational redshift? Get solution
10. Find the relative frequency shift Δf/f for light emitted at the surface of the sun (radius 6.96 × 105 km, mass 1.99 × 1030 kg) if the light is received at (a) the planet Mercury and (b) Earth. Get solution
10q. We mention in the text that gravitational redshifts can be observed and measured during the collapse of a star into a black hole. When might the redshifts cease? Get solution
11. A He-Ne laser with wavelength 632.8 nm is fired from a great distance toward a neutron star with mass 4.5 ×1030 kg and radius 12 km. What is the wavelength of light received at the neutron star’s surface? Get solution
11q. We mentioned that astronauts can tell whether they are in outer space or “falling around Earth” by observing a drop of water in the corner of their spacecraft. What are the tidal forces that were mentioned, and where do they come from? Why are they called “tidal” on Earth? Get solution
12. What is the value of the Schwarzschild radius for the moon? (mmoon = 7.35 × 1022 kg) Get solution
12q. Why can we conclude from Equation (15.1) that the inertial and gravitational masses are equal? ... Get solution
13. Calculate the Schwarzschild radius for Jupiter. (mJupiter = 1.90 × 1027 kg) Get solution
13q. Explain why it’s expected that primordial black holes should not last for a long time before evaporating completely. Why is the last part of such a black hole’s lifetime described in the text as comparable to numerous hydrogen bomb explosions? Get solution
14. Stephen Hawking has predicted the temperature of a black hole of mass M to be ..., where k is Boltzmann’s constant. (a) Calculate the temperature of a black hole with the mass of the sun. Discuss the implications of the temperature you calculate. (b) Find the temperature of a supermassive black hole, which may exist at the center of some galaxies, with a mass 6.0 × 109 times the sun’s mass. Get solution
14q. Some concern was expressed that the high-energy particles produced by the Large Hadron Collider might generate small black holes that could grow out of control and eventually consume all of Earth’s mass. Why is this not a likely scenario? Get solution
15. Calculate the mass and Schwarzschild radius of a black hole at room temperature (see Problem 14). How many solar masses is this? Problem 14 Stephen Hawking has predicted the temperature of a black hole of mass M to be ..., where k is Boltzmann’s constant. (a) Calculate the temperature of a black hole with the mass of the sun. Discuss the implications of the temperature you calculate. (b) Find the temperature of a supermassive black hole, which may exist at the center of some galaxies, with a mass 6.0 × 109 times the sun’s mass. Get solution
16. The supermassive black hole at the center of the NGC 4261 galaxy is thought to have a mass of 1 billion suns. (a) Calculate its Schwarzschild radius and compare it with the size of our solar system. (b) How much time would this black hole take to evaporate by Hawking radiation? Get solution
17. (a) Use the known lifetime of the universe to determine the mass of a black hole that would evaporate all its mass during that time. (b) How likely is it that a black hole of this mass could exist? Get solution
18. Determine the constant α in Equation (15.10). ... Get solution
19. Because the evaporation rate of a black hole increases as the black hole’s size decreases, a small primordial black hole releases energy at a fantastic rate. Find the mass of a black hole that would release energy equivalent to a one-megaton (4.2 × 1015 J) hydrogen bomb every second. Because the evaporation rate of a black hole increases as the black hole’s size decreases, a small primordial black hole releases energy at a fantastic rate. Find the mass of a black hole that would release energy equivalent to a one-megaton (4.2 × 1015 J) hydrogen bomb every second. Get solution
20. For a black hole with the mass of our moon (7.3 × 1022 kg) find (a) its Schwarzschild radius; (b) its effective temperature; and (c) the potential energy associated with this black hole being just above Earth’s surface. Get solution
21g. The Global Positioning System satellites operate at an altitude of 20,200 km and use communication frequencies of 1575.42 MHz. Find the gravitational frequency change with respect to Earth. Get solution
22g. Derive Equation (15.4). ... Get solution
23g. One of the communication frequencies that the International Space Station uses is 259.7 MHz. (a) Find the gravitational frequency change with respect to Earth when the station is at its mean altitude of 352 km. Assume g is constant and equal to 9.80 m/s2. (b) Now do a more precise calculation of the frequency shift, without assuming that g is constant. Get solution
24g. Weightlessness occurs only at the center of mass of the International Space Station as it rotates 350 km above Earth. Calculate the effective g that an astronaut in the station who is 3 m closer to Earth than the center of mass would feel. You may choose to ignore relativistic effects. Get solution
25g. Find the mass of a particle with a Compton wavelength of πrS where rS is the Schwarzschild radius. This mass is called the Planck mass mP, and the energy required to create the mass is called the Planck energy EP = mPc 2. Determine the values of both the Planck mass and energy. Get solution
26g. The length scale on which the quantized nature of gravity should first become evident is called the Planck length. (a) Determine it using dimensional analysis using the fundamental constants G, h, and c. (b) Determine it by finding the de Broglie wavelength of the Planck mass of Problem 25. Are the values close to the value of 10-35 m? Get solution
27g. Use the fundamental constants G, h, and c and dimensional analysis to determine a time constant called the Planck time. How much time would it take light to travel the Planck length discovered in the previous problem? Are these two times consistent? Get solution
28g. A communications satellite is at a geosynchronous orbit position (35,870 km above Earth’s surface) and communicates with Earth at a frequency of 2.0 × 109 Hz. What is the frequency change due to gravity? Get solution
29g. Stephen Hawking’s derivation of the black hole temperature used the fact that the black hole’s entropy is given by ... Complete the derivation using the thermodynamic definition of temperature .... Assume that the black hole’s energy is entirely mass-energy, that is, U = Mc2. Get solution
30g. It is written in the text that light traveling horizontally across the continental United States should fall about 1 mm due to gravity. Determine the approximate vertical fall for light traveling from Los Angeles to New York City. Get solution
