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Unformatted text preview: MA 2930, March 9, 2011 Worksheet 7 Solutions 1. Write u = cos t + √ 3 sin t as R cos( ωt φ ) . How would the amplitude, R , and phase, φ , of u change if you wrote it as R sin( ωt φ ) ? Here is another way to do it: u = cos t + √ 3 sin t = q ( 1) 2 + √ 3 2 [ 1 q ( 1) 2 + √ 3 2 cos t + √ 3 q ( 1) 2 + √ 3 2 sin t ] = 2( 1 2 cos t + √ 3 2 sin t ) = 2[cos(2 π/ 3) cos t + sin(2 π/ 3) sin t ] = 2 cos( t 2 π/ 3) = 2 sin( t 2 π/ 3 + π/ 2) 2. A mass weighing 1 kg stretches a spring 0.1 m. The mass is pushed upward, contract ing the spring a distance of 1 m, and then set in motion with a downward velocity of 1 m/s. (a) Solve for position of the mass at any time t . Determine its frequency, period, amplitude and phase. (You can take g = 10 m/ s 2 .) Suppose the spring constant is k . Then, at rest, mg = kx ⇒ 10 = 0 . 1 k ⇒ k = 100 kg/ s 2 . The differential equation governing the motion of the springmass system is mx 00 + kx = x 00 + 100 x = 0 where x ( t ) is the position of the mass at time t measured from the equilibrium location in the downward direction. The general solution of the system is x ( t ) = c 1 cos 10 t + c 2 sin 10 t where c ’s are to be determined from the initial conditions, which are x (0) = 1 and x (0) = 1. So, 1 = x (0) = c 1 and 1 = x (0) = 10 c 2 . Thus, the position of the mass is given by x ( t ) = cos 10 t + 1 / 10 sin 10 t It’s clear that the angular frequency of motion, ω = 10 rad/s, and period, T = 2 π/ω = π/ 5 s. To determine the amplitude and the phase we have to express the5 s....
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 Spring '07
 TERRELL,R
 Differential Equations, Equations, Sin, Cos

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