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Homework Assignment 5 Physics 55 Made available: Due at my office: Sunday, October 2, 2005 Wednesday, October 5, 2005 Problem 1: Center of Mass for The Sun and Jupiter The position of the center of mass of two masses is given by the formula xCM = (x1 M1 + x2 M1 )/(M1 + M2 ) where M1 and M2 are the masses of the two objects and x1 and x2 are the positions along some ruler connecting the two masses. For example, if you put the Sun at the beginning of the ruler, xSun = 0 and xJupiter would be the distance of Jupiter to the Sun. By looking up in the back of your text the masses of the Sun and of Jupiter, calculate how far away from the center of the Sun is the Jupiter-Sun center of mass (express your answer as a multiple of the radius of the Sun). Recall from our discussion in class that two masses orbit about their center of mass, so the Sun is not sitting still in space but orbiting around the Jupiter-Sun center of mass in a small elliptical orbit. Why is it that the Greeks and others did not notice this fact? Problem 2: Understanding A Neutron Star Large stars at the end of their lives can form a neutron star, which is basically a giant ball of neutrons 10 km in diameter but with the mass of an entire star, about 1.5 times the mass of our Sun, 2 × 10 30 kg. 1. What is the gravitational acceleration at the surface of a neutron star, expressed as a multiple of the Earth’s gravitational acceleration g ≈ 10 m/s2 ? 2. Calculate the escape velocity for a neutron star and express your answer as a fraction of the speed of light c. 3. For extra credit, solve Problem 22 on page 592 of the text: given that a neutron star bumping into the Earth would eventually wrap the entire Earth around its surface as a thin layer of nuclear matter, what would the thickness of that layer be? Problem 3: A Personal Black Hole Assuming that you have a mass of 70 kg (150 lb) and were somehow squeezed into a small enough volume to become a black hole, calculate your personal Schwarzschild radius r = 2GM/c 2 and compare that radius with the radius of a proton, which is about 10−15 m. As aside: There used to be a firm in California that would take the cremated ashes of a beloved one and compress the ashes under high pressure to produce a colored diamond, which could then be put on display or worn as jewelry. The resulting diamond was about a inch in size for a human. State of the art high-pressure diamond anvils can produce enough pressure to turn hydrogen gas into a metal (similar to what is believed to lie at the cores of Jupiter and Saturn) but the human race is far from having the technology to produce black holes by compressing some object. 1 Problem 4: A Tide Problem The average distance from the Moon to the center of the Earth is 384,000 km and the diameter of the Earth is 12,800 km. 1. Calculate the gravitational force that the Moon exerts on a 1-kg rock at the point on the Earth’s surface closest to the Moon. 2. Calculate the gravitational force that the Moon exerts on a 1-kg rock at the point on the Earth’s surface furthest from the Moon. 3. Find the tidal force acting to pull these rocks apart, i.e., the difference between the forces in the previous two parts. (Remember that a force is an arrow that has a direction as well a magnitude so you need to indicate a direction to fully answer this problem). 4. By comparing the magnitude of this tidal force to the rock’s weight on the Earth, explain briefly whether you expect this tidal force to cause a large or small deformation of the Earth. Problem 5: Light Formed By Matter-Antimatter Annihilation A proton p is a positively charged particle with mass 2×10−27 kg while an antiproton p (pronounced “p-bar”) is the antimatter form of a proton, and is a negatively charged particle of identical mass. When a proton and antiproton come sufficiently close to one another, they can disappear (scientists say that they “annihilate one another”) and in their place appear two photons of equal energy, with the total photon energy equal to the total rest-mass energy of the protons. (The rest-mass energy of an object whose mass is m is given by Einstein’s famous formula E = mc2 , see p. 119 of the text. In fact, matter-antimatter annihilation is the only known way to extract all of the rest-mass energy of some object). 1. What is the energy of each of the photons in joules (denoted by the letter J)? 2. Given that an amount of energy of about 10−18 J is enough to break any chemical bond, explain whether or not it would be dangerous for people to be exposed to these photons. 3. What is the wavelength λ and frequency f of the photons produced by this matter-antimatter annihilation? 4. By referring to Figure 6.6 on page 157 of the text, explain what kind of light corresponds to these photons: radio, infrared, ultraviolet, etc.? 5. Extra credit: What could you conclude if, with a telescope, you were to observe a trail of light across part of the sky several degrees wide with this wavelength of light? Problem 6: Optional Extra Credit Problems 1. In the not so far future, it may be possible to land an astronaut on an asteroid. Based on how high you can jump on earth, determine the maximize size of a spherical asteroid that you could jump completely off of. The typical density of a rocky asteroid is about 3,000 kg/m3 . 2. You are enjoying a Caribbean vacation and happen to have a stopwatch with you at the beach. As you watch the Sun set over the ocean, you carry out the following eccentric sequence of events: (1), you lie down on your stomach in the sand and wait until the top of the sun just disappears below the horizon; (2), you then quickly stand up and simultaneously start your stopwatch. By standing up, a bit of the sun is now visible again and (3), you wait until the top of the sun again dips below the horizon, at 2 which point you stop the stopwatch. Knowing this elapsed time, your height, and that a day lasts 24 hours, explain how you can deduce the radius of the Earth. (And next time you find yourself watching a sunset at the beach, give this a try and compare your answer with the known value of 6400 km.) Problem 7: Comments about the Homework and Course • About how long did this assignment take to complete? • Do you feel that you are understanding the course material? If not, please indicate what topics or ideas you would like to understand better. • Other comments or suggestions about the homeworks, lectures, or observation sessions? 3