Read my notes on Boyle’s Law and Viscosity. Do the following questions based on those notes:
1. The density of air at room temperature is about 1.29 kg/m3. Use this together with the definition of 1 atm above to find the constant C in the Law of Atmospheres written above. Use you result to estimate the atmospheric pressure on top of the Blue Ridge (say 4,000 feet), Snowmass (11,000 feet) and Mount Everest (29,000 feet).
2. Suppose a mass m of fluid moving at v1 in the x-direction mixes with a mass m moving at v2 in the x-direction. Momentum conservation tells us that the mixed mass 2m moves at ½( v1 + v2). Prove that the total kinetic energy has decreased if v1, v2 are unequal.
3. Plot on a graph the velocity profiles for two rivers, one of depth h and one 2h, having the same values of r, g, q and h. What is the ratio of the surface velocities of the two rivers? Suppose that one meter below the surface of one of the rivers, the water is flowing 0.5 m×sec-1 slower than it is flowing at the surface. Would that also be true of the other river?
4. 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.
5. 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.)