A force of 3 N stretches a spring by 1 m. (a) Find the spring constant k. (b) A

Question

A force of 3 N stretches a spring by 1 m. (a) Find the spring constant k. (b) A mass of 4 kg is attached to the spring. At t-0, the mass is pulled down a distance 1 m from equilibrium and released with a downward velocity of 0.5 m/s. Assuming that damping is negligible, determine an expression for the position y of the mass at time t. (Assume that y is the distance in meters from the equilibrium position, where distances are measured positive in the downward direction. Enter your answer as an equation involving y andt.) V 3 Find the circular frequency of the system and the amplitude, phase, and period of the motion. circular frequency-1 X s-1 amplitude phase 6 period

Explanation / Answer

b] y= A sin(wt+ phi) where w = sqrt(k/m) = sqrt(3/4) = 0.866

y = A sin (0.866t + phi)

v = A*0.866* cos(0.866t + phi)

at t=0, dividing first equation by second,

y/v = 1/0.866 tan phi

-1/-0.5 = 1.1547 tan phi

phi = arctan(2/1.1547) = -2.094 rad

A = y/sin phi = -1/sin -2.094 = 1.15

y = 1.155 sin (0.866t -2.094 )

   circular frequency w = 0.866 s^-1

phase = -2.094

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