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Physics 109n Homework # 9

Due 2:00, Tuesday November 12, 1996

1. Draw a diagram showing the orbits of earth, Jupiter and Io and explain in your own words how Römer found the speed of light by observing eclipses of Io.

2. Explain why (in the days before long distance communication) it was necessary to have some kind of an accurate clock to find where you were on earth - your longitude - and how the satellites of Jupiter could be used as a clock.

3. The four biggest satellites of Jupiter, those found by Galileo, are named after four Greek characters: Callisto, Europa, Ganymede and Io. Who were these people?

4. State Newton's Second Law.

Suppose you are standing on a spring bathroom scale in an elevator which is at rest. What is your acceleration? What forces are acting on you? What is the reading on the bathroom scale?

Suppose now you push the button for a higher floor, and the elevator begins to move. After three seconds it is going up at a steady one meter per second. Did you accelerate? Are you still accelerating? How do you think the reading on the scale will vary as the elevator begins to move then settles to a steady speed? (Just answer qualitatively - no numbers expected.)

Now imagine disaster - the rope holding the elevator snaps, there are no safety features, and the elevator plunges down the shaft in free fall. You avail yourself of this golden opportunity to check Newton's Laws further, since it is unlikely to be repeated, at least for you personally. You stay calmly on the spring scale. What does it read?

Read the second selection from Two New Sciences I've put on the Web until you find a sentence or two relevant to the above disaster scenario, and explain the connection.

5. We've weighed the earth, let's weigh the sun.

We know from Cavendish's experiment, and Newton's Universal Law of Gravitation, that the force of attraction between the earth and the sun is GMm/r2, where G is the gravitational constant, found by Cavendish to be 6.67 x 10-11, M is the mass of the sun, m the mass of the earth, and r the distance of the earth from the sun, which is 150,000,000 kilometers. (NOTE: in the formula for the force, r must be given in meters).

(This force causes the earth to accelerate towards the sun, that is, it deviates from straight line motion into a circle, just like Newton's cannonball. The strategy is to find how far it falls below a straight line in one second and figure out from that what its acceleration towards the sun must be. This acceleration is caused by the gravitational attraction force, which depends on the mass of the sun. Newton's Second Law gives the relation between the acceleration and the force, and enables us to find the mass of the sun.)

Using the fact that the earth goes around its orbit completely in one year, find how far the earth travels in one second. Now, find how far it "falls" below straight line motion in that one second. (HINT: call the distance it falls x, then write down Pythagoras' theorem for the usual triangle, and argue that x2 can be safely neglected. Then it's easy to find x.)

Since the time interval we are taking is just one second, the distance the earth falls below the straight line in that period must be equal to its average velocity in that direction (that is, towards the sun) during the one second. So what is the earth's velocity in that direction at the end of the one second period?

So what is its acceleration?

Write down Newton's Second Law for this acceleration, which is caused by the sun's gravitational attraction, and from it deduce the mass of the sun. Notice that you do not need to know the mass of the earth - why not?