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Unformatted text preview: Lecture 8: Constraints (21 Sep 09) A. Falling body 1. Vector directions using spherical polar coordinates and assuming spher ical earth ˆ r, ˆ ϑ, ˆ ϕ , ~ω = ω ˆ z . Cross products: ˆ r × ˆ ϑ = ˆ ϕ ; ˆ ϑ × ˆ ϕ = ˆ r ; ˆ ϕ × ˆ r = ˆ ϑ cyclic 2. For the angular velocity ~ω = ω ˆ z , use ˆ z = cos ϑ ˆ r sin ϑ ˆ ϑ and then ˆ z × ˆ r = sin ϑ ˆ ϕ 3. Hence the centrifugal term is (correct typo) ~ω × ( ~ω × ˆ r ) = ω 2 [cos ϑ sin ϑ ˆ ϑ + sin 2 ϑ ˆ r ] and the effective acceleration at earth’s surface is ~g = g ˆ r ~ω × ( ~ω × R e ˆ r ) = [ g ω 2 R e sin 2 ϑ ]ˆ r + 1 2 ω 2 R e sin 2 ϑ ˆ ϑ 4. Drop the subscript for bodyframe time derivatives, drop the centrifugal term and approximate the gravitational acceleration as ~g = g ˆ r ; hence m d 2 r dt 2 = m~g 2 m~ω × d r dt 5. Solve in successive approximations: r ( t ) = r ( t ) + r 1 ( t ) + ... (order in powers of ω ) d 2 r dt 2 = g ˆ r d 2 r 1 dt 2 = 2 ~ω × d r dt 6. Initial condition is static: r (0) = ( R e + h )ˆ r and ˙ r (0) = 0: ⇒ r ( t ) = [ R e + h 1 2 gt 2 ]ˆ r ; the time of fall is t f = q 2 h /g ....
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This note was uploaded on 09/29/2009 for the course PHYS 711 taught by Professor Bruch during the Fall '09 term at University of Wisconsin.
 Fall '09
 BRUCH

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