a01solutions - PHYS 218 SOLUTION TO ASSIGNMENT 1 02 Feb...

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PHYS 218 - SOLUTION TO ASSIGNMENT 1 02 Feb 2007 By Chung Koo Kim e-mail : [email protected] 1. 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 l = l 0 + mg k (c) If the mass is pulled down by y from the new equilibrium, we get, from the result of (b), F = m ¨ y = - k ( l + y - l 0 ) + mg = - ky thus the equation of motion has the same form. Then what is different in the presence of gravity? Nothing except for the center of oscillation. Center position, is incorporated as the constant term of general solution (C in y ( t ) = A sin( ωt )+C), and it is determined by the equilibrium of force. Neither amplitude nor frequency change. 2. Pendulum (a) Eq. of motion can be found using either force or torque. Using the latter, - mglsinθ = ml 2 ¨ θ ¨ θ = - g l sinθ ∼ - g l θ ω 0 = g l , T = 2 π l g (b) From the result of (a), for l 20 m , T 9 . 0 s (c) If we assume that the number of pulse per minute is roughly 60 (or N in general) times, the actual time interval between each pulse is 1 sec (or 60/N). Therefore if Galileo measured the period by counting his pulse, the precision would be order of one interval. He could have improved the precision by counting his pulse over many periods and divide by total number of oscillation. 1
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(d) The amplitude of a damped harmonic oscillator with a damping constant γ is given by A ( t ) = A 0 e - γt/ 2 from the general solution θ ( t ) = A 0 e - γt/ 2 cos( ω 0 t + φ ). So if the amplitude decreased by a factor of 2 in 30min, we solve A 0 / 2 = A 0 e - γt/ 2 for γ to get γ = 2 ln 2 t = 7 . 7 × 10 - 4 s - 1 and Q = ω 0 γ = t 2 ln 2 g l 909 (e) For Q = 10 10 , the damping constant γ is : γ = ω 0 Q = 7 × 10 - 11 s - 1 Then, from the amplitude relation, we get A 2006 = A 1581 e - γt/ 2 0 . 627 A 1581 , finding that the amplitude is still about 62.7% of the original one even after 425 years have passed. Here we see that higher Q yields less damping, or higher stability of operation.
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