- Find the length of the cardioid, r = 1 + cosθ,θ ∈ . Hint: A parametrization is x=cos θ,y=sin θ.
- In general, show that the length of the curve given in polar coordinates by
r = f,θ ∈equals ∫
_{a}^{b}dθ. - Using the above problem, find the lengths of graphs of the following polar
curves.
- r = θ,θ ∈
- r = 2cosθ,θ ∈
- r = 1 + sinθ,θ ∈
- r = e
^{θ},θ ∈ - r = θ + 1,θ ∈

- r = θ,θ ∈
- Suppose the curve given in polar coordinates by r = ffor θ ∈is rotated about the y axis. Find a formula for the resulting surface of revolution. You should get
- Using the result of the above problem, find the area of the surfaces obtained by
revolving the polar graphs about the y axis.
- r = θ sec,θ ∈
- r = 2cosθ,θ ∈
- r = e
^{θ},θ ∈ - r = sec,θ ∈

- r = θ sec
- Suppose an object moves in such a way that r
^{2}θ^{′}is a constant. Show that the only force acting on the object is a central force. - Explain why low pressure areas rotate counter clockwise in the Northern hemisphere and clockwise in the Southern hemisphere. Hint: Note that from the point of view of an observer fixed in space above the North pole, the low pressure area already has a counter clockwise rotation because of the rotation of the earth and its spherical shape. Now consider 17.2. In the low pressure area stuff will move toward the center so r gets smaller. How are things different in the Southern hemisphere?
- What are some physical assumptions which are made in the above derivation of Kepler’s laws from Newton’s laws of motion?
- The orbit of the earth is pretty nearly circular and the distance from
the sun to the earth is about 149 × 10
^{6}kilometers. Using 17.16 and the above value of the universal gravitation constant, determine the mass of the sun. The earth goes around it in 365 days. (Actually it is 365.256 days.) - It is desired to place a satellite above the equator of the earth which will rotate
about the center of mass of the earth every 24 hours. Is it necessary that
the orbit be circular? What if you want the satellite to stay above the
same point on the earth at all times? If the orbit is to be circular and
the satellite is to stay above the same point, 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 × 10
^{24}kilograms. Such a satellite is called geosynchronous. - Show directly that the area of the inside of an ellipse += 1 is πab. Hint: Solve for y and consider the top half of the ellipse.
- Recall the formula derived above for the angular velocity vector
In the case of the rotating earth,

^{∗},j^{∗},k^{∗}. Show directly that Ω= ωk^{∗}as claimed above. - Suppose you have
(17.25) and x

= x^{′}= y= y^{′}= 0 . Show that x= y= 0 . Show this implies there is only one solution to the initial value problem 17.23 and 17.24. Hint: If you had two solutions to 17.23 and 17.24,,ỹ and,ŷ, consider x =−and y = ŷ −ỹ and show x,y satisfies 17.25. To show the first part, multiply the first equation by x^{′}the second by y^{′}add and obtain the following using the product rule.Thus the inside is a constant. From the initial condition, this constant can only be 0.

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