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Q6_1130 - MATH 415.01 QUIZ 6A SHOW ALL YOUR WORK GOOD LUCK...

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Unformatted text preview: MATH 415.01 QUIZ 6A Feb 17, 2010 SHOW ALL YOUR WORK! GOOD LUCK! NAME: 1. (6.5 pts.) A mass that weighs 8 lb stretches a spring 6 in. The system is acted on by an external force of 8sin(8 t ) lb. If the mass is pulled down 3 in. and then released, determine the position of the mass at any time. SOLUTION: m = 8 / 32 = 1 / 4, γ = 0, k = 8 / (6 / 12) = 16. Also, we have u (0) = 3 / 12 = 1 / 4 , u (0) = 0 i.e. The equation is: 1 / 4 u 00 + 16 u = 8sin(8 t ). Characteristic equation 1 / 4 r 2 + 16 = 0. So, r 2 =- 64. Then u c = c 1 cos(8 t ) + c 2 sin(8 t ). Find particular solution: U ( t ) = ( A cos(8 t ) + B sin(8 t )) t , U ( t ) = A cos(8 t ) + B sin(8 t ) + t (- 8 A sin(8 t ) + 8 B cos(8 t )) and U 00 ( t ) =- 16 A sin(8 t ) + 16 B cos(8 t ) + t (- 64 A cos(8 t )- 64 B sin(8 t )) Then, solve for A and B. 1 / 4 U 00 + 16 U = 8sin(8 t ) i.e.- 4 A sin(8 t ) + 4 B cos(8 t ) + t (- 16 A cos(8 t )- 16 B sin(8 t )) + 16( A cos(8 t ) + B sin(8 t )) t = 8sin(8 t ) i.e....
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Q6_1130 - MATH 415.01 QUIZ 6A SHOW ALL YOUR WORK GOOD LUCK...

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