The solution to a vibrating spring problem is x(t) = 6 cos 21 + 8 sin 2t. What i
ID: 3142553 • Letter: T
Question
The solution to a vibrating spring problem is x(t) = 6 cos 21 + 8 sin 2t. What is the amplitude of the spring's motion? Show work. b. What is this spring's period? c. Write this spring's motion as a single sine function (x(t) = A sin(omega t + phi)), showing the phase shift clearly. d. What was the initial position and velocity of the weight at t = 0 seconds? Write both the numbers and a description of what each means in the spring situation. e. How long does it take for the weight to pass through the equilibrium position for the first time? Give the exact amount of time (no decimals) as well as an approximate (decimal) value for the time. Show clear work to support your answer.Explanation / Answer
As per chegg policy, I wil answer only first 4 subparts.
a)
Amplitude = ( 62 + 82 )1/2 = 10
b)
x = 6 cos 2t + 8 sin 2t
Speed of the spring is obtained by differentiating the x
dx/dt = -12 sin 2t + 16 cos 2t
c)
x = 6 cos 2t + 8 sin 2t
x = A sin (wt + f) {f is representing phi}
A = ( 62 + 82 )1/2 = 10
w = 2
f = pi/2 - arctan (6/8) =36.8698976 degrees
x = 10 sin (2t + 36.8698976)
phase shift is 36.8698976
d)
at t = 0
x (t) = 6 cos 2t + 8 sin 2t
x (0) = 6 units
velocity (t) = -12 sin 2t + 16 cos 2t
velocity (0) = 16 units