MTH 444/544 Homework #1

NOTE: To see greek characters in this assignment rendered properly, it is recommended that you use a standards-compliant browser such as Firefox. (IE does not work properly.)

1. Consider a motionless "perfect gas" at constant temperature in a uniform gravitational field of acceleration, g. For a perfect gas, the density is proportional to the pressure: let's say &rho = a p.
(a) Using the balance of momentum formula we derived in class, obtain expressions for the density, &rho(z) and pressure, p(z). (z is the coordinate parallel to the gravitational field.) Call the pressure at z = 0 p₀.
(b) Using your knowledge of the density and pressure of air in our classroom, and looking up the radius of the Earth on Google or elsewhere, make a crude estimate of the total mass of the Earth's atmosphere, by assuming the atmosphere is a perfect gas at constant temperature, and doing an integral. (Note: you may simplify your calculation by anticipating that the atmosphere is very thin compared to the Earth radius.)

(c) Estimate the “thickness” of the atmosphere under the same assumptions. Since in the above model the atmosphere technically extends to infinity, you will have to define the thickness in some appropriate way. Explain and justify your definition.

2. (a) Write down an expression for the pressure, p, in a motionless incompressible fluid of uniform density &rho, in a uniform gravitational field of acceleration, g.

(b.ii) What would be the height of the column if the liquid were mercury which is about 13.6 times denser than water?
(c) See the "Barometer" item on this Weather page. Write down what it says there and relate that information to your answer to part (b).

3. Surface of spinning fluid.
Obtain a theoretical prediction of h1-h0 in our rotating bucket experiment. The center goes down and the outer edge changes from h0 up to h1. We derived a formula for the shape. You also know that the total volume of the fluid is the same when spinning and when stationary. The rotation rate was 45rpm. The diameter of the (approximately cylindrical) bucket was 26cm.