Chapter #15 Solutions - Modern Physics for Scientists and Engineers - Andrew Rex, Stephen T. Thornton - 3rd Edition

 

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 400 km above the 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 about ... kg and a radius of about 10 km. 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 550 nm (the middle of the visible range) that is emitted from the sun. 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

9q. Why is it difficult to test models of general relativity experimentally? Get solution

10. What is the value of the Schwarzschild radius for the moon? (mmoon = 7.35 × 1022 kg) 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. Calculate the Schwarzschild radius for Jupiter. (mJupiter = 1.90 × 1027 kg) 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. Stephen Hawking has predicted the temperature of a black hole to be ...is Boltz-mann's constant. Calculate the temperature of a black hole with the mass of the sun. Get solution

12q. Why can we conclude from Equation (15.1) that the inertial and gravitational masses are equal? ... Get solution

13. 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

14. 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

15. (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

16. Determine the constant α in Equation (15.10). ... Get solution

17g. 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

18g. Derive Equation (15.4). ... Get solution

19g. One of the communication frequencies that the space shuttle orbiter uses is 294 MHz. Find the gravitational frequency change with respect to Earth when the or-biter is at an altitude of 300 km. Assume g is constant. Get solution

20g. Weightlessness occurs only at the center of mass of the space shuttle orbiter as it rotates 300 km above the Earth. Calculate the effective g that an astronaut in the orbiter would feel who is 3m closer to the Earth than the center of mass. You may choose to ignore relativistic effects. Get solution

21g. 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

22g. 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

23g. 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

24g. 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


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