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374bhw2 - HW#2 Phys374Spring 2007 Due before class Friday...

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HW#2 —Phys374—Spring 2007 Prof. Ted Jacobson Due before class, Friday, Feb. 9 Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/374b/ [email protected] 1. The Newtonian gravitational potential energy of a point mass m located at radius r in the gravitational field of a spherically symmetric mass M is ϕ ( r ) = - GMm/r . For the case where M is the earth, it is natural to make a Taylor expansion of ϕ ( r ) about the radius of the earth r e . (a) Work out the first three terms in the Taylor series for ϕ ( r ) about r e in powers of h = r - r e . (b) Identify the “ mgh ” term and give the gravitational acceleration g in terms of G , M , and r e . (c) What is the order of magnitude of the ratio of the O ( h 2 ) term to the O ( h ) term if h is (i) one meter, (ii) one kilometer, (iii) 350 kilometers (international space station altitude), (iv) 6 . 6 earth radii (geosynchronous orbit)? In which cases does the O ( h ) term give a decent approximation? [5+2+3 pts.] 2. Consider the cubic equation ay 3 + y + 2 = 0, with a > 0. (a) Display the location of the real roots graphically, by sketching the graphs of y +2 and - ay 3 (put y on the horizontal axis) and seeing where they intersect. Show in your sketch three cases: a is small, equal to, and large compared to 1. [2 pts.] (b) Determine the leading order a dependence of the roots in the limits (i) a 1 and (ii) a 1. (Don’t solve it exactly, even if you can.) [8 pts.] 3. Relativistic Energy The total energy of a free particle of mass m and speed v in special relativity is E = mc 2

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374bhw2 - HW#2 Phys374Spring 2007 Due before class Friday...

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