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.
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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
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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?
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