1.    We notice that Uranus takes 2 hours, 6 minutes, and 40 seconds to occult (pass in front of) a certain star. We have long known that Uranus travels across the sky at an average speed of 0.0005 seconds of arc in one second of time.

a) Compute the angular diameter of Uranus, in seconds of arc. ONLY USE THE INFORMATION GIVEN ABOVE!

You will not use the angular size formula for this part! All you need to know is in the problem statement. As Uranus passes in front of the star, the star appears to be in motion behind Uranus. So we can say that the star appears to travel behind Uranus at the stated speed, for the stated amount of time. How many seconds of arc in distance does the star travel behind Uranus, or Uranus travel in front of the star?

Look at it this way: say I want to drive from Austin to Dallas. The trip takes a total of four hours, if my average speed is 50 miles per hour. Given this information, what is the distance from Austin to Dallas? How did you compute that? You will follow the exact same procedure to compute the distance of Uranus' "trip" across the star. This is the angular size of Uranus.

The angular size of Uranus should only be a few seconds of arc.

b) Look up the average distance of Uranus in the book, and then compute the actual diameter of Uranus, using the angular size you computed in part a).

Now you use the angular size formula! This will be very similar to Example 1 in that handout. Use the average distance between Uranus and the Sun, since you don't know how Uranus and Earth are situated.

You can confirm your answer in the textbook. Don't feel bad if the answer doesn't exactly match! Just get close!

2.     Use the information given for Callisto and Newton's Law of Gravity to compare the gravity felt by a person on the surface of Callisto to the gravity felt by a person on the surface of Earth. How much would a person who tips the scales at 200 pounds on Earth "weigh" on Callisto?

HINT: This is a straightforward application of the weight equation. Look at Example 1 of the weight handout and the answers to Homework 2 for models. The process is the same; only the numbers are different!

Do you think you could jump free of Callisto's gravity using just your legs? Explain.

Think about it this way: you have just computed how much a 200-pound person on Earth would weigh on Callisto. Say to yourself: "200 pounds on Callisto feels like X pounds here on Earth." Could you launch X pounds into orbit here on Earth using just your legs? That's the same as launching 200 pounds on Callisto.

Make a non-mathematical statement about how Callisto's gravity compares to the Moon's. Look at Callisto's density compared to the Moon's and try to come up with a non-mathematical explanation why the two compare that way.

HINT: Focus on compositions, what the two planets are made of, and how that affects their gravitational pulls.

3.     Use the orbital information given for Io and Newton's Version of Kepler's Third Law to compute the mass of Jupiter, first in units of solar masses, then in grams. Notice that none of the figures in the table are in the proper units, so you must convert all! Be as accurate as you can with your figures. Compare your final answer to that given in the text. For two BONUS points, prove that you don't need to know the mass of Io to do this problem.

HINT: You've already worked with Newton's Version of Kepler's Third Law on Example 2 of the Newton's Version of Kepler's Third Law handout and Homework 2. Remember, the procedure is the same; only the numbers are different!

You can get the mass of Io from the text, but it is given in terms of the mass of the Moon, for some reason. You must convert this first to grams (using the mass of the Moon), then to solar masses (using the mass of the Sun). Remember that we are concerned only with Jupiter and Io, so the distance between Jupiter and the Sun, for example, is totally irrelevant!

For bonus points, you can prove that the mass of Io is not needed. How you do it is up to you, but there are at least two good ways.

Updated 7/5/06

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