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**Unformatted text preview: **PHY 3221: Mechanics I Fall Term 2010 Midterm Exam 2, October 27, 2009 • This is a closed book exam lasting 50 minutes. • Since calculators are not allowed on this test, if the problem asks for a numerical answer, answering 2 + 2 is as good as 4, and √ 2 is as good as 1 . 4142 ... . • There are four problems worth a total of 20 pts. All four problems appear on the second page of this test. Begin each problem on a fresh sheet of paper. Use only one side of the paper. Avoid microscopic handwriting. • Put your name, the problem number and the page number in the upper right-hand corner of each sheet. • To receive partial credit you must explain what you are doing. Carefully labelled figures are important! Randomly scrawled equations are not helpful. • Draw a box around important results (or at least results which you think might be im- portant). • Good luck! 1 Problem 1. [4 pts] In Einstein’s theory of general relativity the Schwarzschild radius R s is the radius at which a spherical body (such as a star or a planet) would become a black hole. For example, if you take a star of mass M and squeeze all of its matter within a sphere of radius R s (or smaller), the gravitational pull on the surface becomes so strong that even light cannot escape from it, and the star becomes a black hole. Knowing nothing about general relativity, use dimensional analysis to derive a formula for the Schwarzschild radius R s in terms of the mass of the star M , Newton’s gravitational constant G and the speed of light c . Solution. First figure out the units of the relevant quantities: [ R s ] = L, [ M ] = M, [ G ] = L 3 M- 1 T- 2 , [ c ] = LT- 1 . Then write R s ∼ M α G β c γ from where [ R s ] = [ M ] α [ G ] β [ c ] γ L = M α L 3 β M- β T- 2 β L γ T- γ giving three equations for α , β and γ : 1 = 3 β + γ, 0 = α − β, 0 = − 2 β − γ....

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