Due Wednesday February 26, 9:00am.
Write and sign the pledge! This means that you are pledging not to discuss the content of this homework with each other, or with anyone else. You are allowed to use books or internet resources, but must acknowledge in writing any which have been of significant help (not counting my notes or the textbook). You can e-mail questions to me—I might or might not answer. If I do, your question plus my answer will go to the whole class.
1. In a car going down a highway at a steady velocity, a
child has a helium-filled balloon on a string, the balloon is at rest directly
above the child, not touching the roof of the car. Now the driver accelerates.
How does the balloon move?
Explain your reasoning.
2. Suppose a satellite is in low earth orbit, that is, in a circular orbit at a height of 200 km., so the radius of the circle is 6600 km., say. We want to raise it to a circular orbit of twice that radius (so it will now be going in a circle at a height of 6800 km above the earth’s surface.)
The technique is to give it two quick boosts: boost1 puts it into an elliptical orbit, where its furthest point from the earth’s center is exactly twice its distance of closest approach, boost2, delivered at the topmost point of the orbit, transfers it to a circular orbit at that radius.
Use conservation of angular momentum and energy in the elliptical orbit to answer these two questions:
(a) By what percentage did boost1 increase its speed?
(b) By what percentage did boost2 increase its speed?
(You may find it helpful in visualizing the intermediate orbit to use the planet spreadsheet, but it is not necessary to do so.)
3. (a) What physical parameters does the period of a pendulum depend on? Write out a dimensional argument to find how the period varies with length.
(b) Assume the planets move in circular orbits around the sun. What physical parameters can the period of the orbit depend on? Use a dimensional argument to establish Kepler’s Law relating period to distance from the sun.
(c) Prove using a dimensional argument that the flow rate in the middle of a wide smoothly flowing river of uniform depth d goes as d3.
4. After a storm, some rivers and lakes become muddy. Assume the mud particles have the density of ordinary rock, say 2500 kg per cubic meter, and assume they are spherical to a good approximation. If the still water in a lake one meter deep takes two days for the mud to settle to the bottom, use Stokes’ Law to give a ballpark estimate of the size of the mud particles.
Problems from Halliday: Chapter 14: 54P. Chapter 15: 16P, 38P, 56P.