This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MA 2930, March 9, 2011 Worksheet 7 Solutions 1. Write u = cos t + 3 sin t as R cos( t ) . How would the amplitude, R , and phase, , of u change if you wrote it as R sin( t ) ? 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 / 3) cos t + sin(2 / 3) sin t ] = 2 cos( t 2 / 3) = 2 sin( t 2 / 3 + / 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 Its clear that the angular frequency of motion, = 10 rad/s, and period, T = 2 / = / 5 s. To determine the amplitude and the phase we have to express the5 s....
View
Full
Document
 Spring '07
 TERRELL,R
 Differential Equations, Equations

Click to edit the document details