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ASTRONOMY 9: HISTORY OF COSMOLOGY

Assignment #8

2000 February 11

Note: Monday 2/14 will be a guest lecture by Professor Ivan King; Wednesday 2/16 lecture has been rescheduled for Thursday 2/17, in 501 Campbell Hall (next to LeConte). You may attend either at 10:00, 11:00, or one additional time to be announced.

Reading Assignment

Course Reader, vol 1.

Readings for 2/17 and 2/18:

Optional Readings (not required)

Journal Homework

Due: February 21 (Monday), at the beginning of class
Note: For numerical problems, you must show your work; simply writing down a correct answer will get no credit. Some of these problems are designed to remind you how to manipulate large numbers (scientific notation) and to practice unit conversions, skills we will need later in the course. Make sure your numbers always have units on them where they are supposed to. You will need a calculator with trigonometric functions. See the instructor for help if these are too unfamiliar! For non-numerical problems, a couple of sentences should suffice.

1.
How fast is a point at the equator moving due to the earth's rotation? Earth's radius is about 6400 km. Convert your answer to miles per hour, and compare to the speed of a typical airplane (500 mph).
2.
Commander Ivanova lands on Z'ha'dum. Before her meeting with Zathras, she has a little time on her hands, so she decides to use Eratosthenes' method to measure the size of Z'ha'dum. At her landing point, she finds that the ``sun'' is directly overhead at noon and casts no shadow. She then walks 1000 km north (ok, she has a lot of time on her hands) and finds that her vertical 1-meter stick casts a shadow of length 32.5 centimeters (cm) at noon. What is the circumference of Z'ha'dum? What is its radius? Compare with the circumference of Earth (express your comparison as a ratio). Hint: Draw some pictures!
3.
How long would it take you to walk all the way around the equator of the Earth? Assume there is no water to get in the way, and that you can walk continuously at a brisk pace of 5 km per hour. Convert your answer to days.

4.
Explain why Aristotle believed there is a ``fifth element''.

5.
What were Aristotle's reasons for believing in a spherical Earth? In what crucial way did his reasons differ from Plato's?

6.
In his work The Sand Reckoner, Archimedes (287-212 BC, famous for running naked through the streets after discovering the principle of buoyancy) computed that 1063 grains of sand would fill the known universe. Assume that each grain of sand has 1024 atoms, and each atom has a mass $1.6\times 10^{-27}$ kilograms (kg).
(a)
How many atoms are there in all the sand?
(b)
What is the mass of a grain of sand?
(c)
What is the total mass of all the sand?
Note: the universe is really much bigger than this!

7.
In class we discussed how Aristarchus estimated that the Sun is about 20 times farther away than the Moon. He did this by estimating that the Moon was half-full (``first quarter'') when it was separated from the Sun by an angle of $87^\circ$.
(a)
It turns out that he was wrong by another factor of 20; the Sun is really about 400 times farther than the Moon. Assuming that the Moon is in a uniform circular orbit (not a very good assumption for this problem, but never mind), what is the true angle between Moon and Sun at first quarter? Hint: Draw a picture!
(b)
By how much was Aristarchus in error about the angle? Express this error as a percentage.
(c)
Is the angle in the error large compared to the error in the distance? What does this imply about the accuracy with which you would have to measure the angle in order to get an accurate measurement of the distance ratio? Does this seem like a good method?

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Up: Astronomy 9 Assignments
jonathan baker
2000-02-11