24.8. EXERCISES 463

(b) r = 2cosθ , θ ∈ [−π/2,π/2]

(c) r = 1+ sinθ , θ ∈ [0,π/4]

(d) r = eθ , θ ∈ [0,2]

(e) r = θ +1, θ ∈ [0,1]

4. Suppose the curve given in polar coordinates by r = f (θ) for θ ∈ [a,b] is rotatedabout the y axis. Find a formula for the resulting surface of revolution. You shouldget

2π

∫ b

af (θ)cos(θ)

√f ′ (θ)2 + f (θ)2dθ

5. Using the result of the above problem, find the area of the surfaces obtained byrevolving the polar graphs about the y axis.

(a) r = θ sec(θ) , θ ∈ [0,2]

(b) r = 2cosθ , θ ∈ [−π/2,π/2]

(c) r = eθ , θ ∈ [0,2]

(d) r = (1+θ)sec(θ) , θ ∈ [0,1]

6. Suppose an object moves in such a way that r2θ′ is a constant. Show that the only

force acting on the object is a central force.

7. Explain why low pressure areas rotate counter clockwise in the Northern hemisphereand clockwise in the Southern hemisphere. Hint: Note that from the point of viewof an observer fixed in space above the North pole, the low pressure area alreadyhas a counter clockwise rotation because of the rotation of the earth and its sphericalshape. Now consider 24.5. In the low pressure area stuff will move toward the centerso r gets smaller. How are things different in the Southern hemisphere?

8. What are some physical assumptions which are made in the above derivation of Ke-pler’s laws from Newton’s laws of motion?

9. The orbit of the earth is pretty nearly circular and the distance from the sun to theearth is about 149×106 kilometers. Using 24.19 and the above value of the universalgravitation constant, determine the mass of the sun. The earth goes around it in 365days. (Actually it is 365.256 days.)

10. It is desired to place a satellite above the equator of the earth which will rotate aboutthe center of mass of the earth every 24 hours. Is it necessary that the orbit becircular? What if you want the satellite to stay above the same point on the earthat all times? If the orbit is to be circular and the satellite is to stay above the samepoint, at what distance from the center of mass of the earth should the satellite be?You may use that the mass of the earth is 5.98× 1024 kilograms. Such a satellite iscalled geosynchronous.

11. Show directly that the area of the inside of an ellipse x2

a2 +y2

b2 = 1 is πab. Hint: Solvefor y and consider the top half of the ellipse.