Physics218sol1

# Physics218sol1 - PHYS 218 SOLUTION TO ASSIGNMENT 1 02 Feb...

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PHYS 218 - SOLUTION TO ASSIGNMENT 1 02 Feb 2007 By Eliot Kapit and Chung Koo Kim e-mail : [email protected] and [email protected] 1. Damped Harmonic Motion (a) Quite obviously, this is a damped pendulum; the factor of exp - 0 . 25 t indicates that the oscillation will decay with time. (b) See graph: (c) We can extract the damping, γ = 2 * 0 . 25 = 0 . 5 s - 1 , the oscillation frequency ω = 12 πs - 1 , the natural frequency ω 0 = ω 2 + γ 2 / 4 = 37 . 7 s - 1 and the initial phase shift α = 3. One cannot measure the mass since the acceleration from gravity is independent of it, and terms such as φ and F 0 all depend on the existence of a driving force, which is not present in this problem. (d) T = 2 π/ω = 1 / 6. The other characteristic time is the decay constant τ = 1 / 0 . 25 = 4 s , which is the time it would take for the amplitude to decay by a factor of 1 /e . 2. Effect of gravity on a hanging spring (a) One arrow pointing down, and another one up, representing weight (= mg ) and restoring force of spring ( = k ( l - l 0 )), respectively. (Diagram omitted) (b) From Newton’s 2nd law for vertical direction, mg - k ( l - l 0 ) = 0, or

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