differential equations - I need the work so i can study this for an test A 3 - k
ID: 2078401 • Letter: D
Question
differential equations - I need the work so i can study this for an test
A 3 - kg mass is attached to a spring with stiffness k = 75 N/m. The mass is displaced 1/4 m to the left and given a velocity of 1 m/s to the right. The damping force is negligible. a. Find the equation of the motion of the mass in the form y = A sin(omega t + phi) b. Give the amplitude, period, and frequency. c. How long after release does the mass pass through the equilibrium position? d. When will it attain the maximum displacement to the right?Explanation / Answer
A) by conservation of energy,
0.5 kx^2 +0.5mv^2 =0.5 kA^2
75*0.25^2 +3*1^2 = 75A^2
A = sqrt (0.25^2 + 3/75)
= 0.32 m
W = sqrt(k/m) = sqrt(75/3) = 5 rad/s
at t=0, y = A sin phi
phi = arcsin (-0.25/0.32) = - 0.897 rad
Hence we have y =0.32 sin(5t - 0.897)
B) Amplitude = 0.32m
Period = 2pi/5 = 1.256 s
Frequency = 5/2pi =0.796 Hz
C) 0 = 0.32 sin (5t - 0.897)
5t - 0.897 = 0
t = (0.897)/5 = 0.18 s
D) here, 0.32 = 0.32 sin(5t-0.897)
5t - 0.897 = pi/2
t = (pi/2 + 0.897)/5
= 0.494 s
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