differential equations A 64 pound weight is attached to the lower end of a coile
ID: 2308775 • Letter: D
Question
differential equations
A 64 pound weight is attached to the lower end of a coiled spring suspended from the ceiling. The weight comes to rest at its equilibrium position, thereby stretching the spring 8 feet. At timet 0, the weight is positioned v3 feet below equilibrium and given a upward velocity of 2 feet per second. (a) Determine the equation of motion of the weight as a function of time. (b) Find the amplitude of the motion. (c) Find the period of the motion. (d) Find the phase angle. (e) Determine the times that the weight passes through equilibrium. (f Determine the second time the weight passes through equilibrium moving upward. (g) Determine the third time the weight passes through equilibrium moving downward. (h) Determine the times that the weight is 1 foot above equilibrium. (i Determine the second time the weight is 1 foot above equilibrium and moving downward. Determine the first time the weight is 1 foot above equilibrium and moving upward.Explanation / Answer
m = 64 pound =29.03 kg
x = 8 x 0.3048 m = 2.44 m
at equilibrium position,
mg -kx = 0
k = (29.03 x 9.8)/2.44 = 116.67 N/m
for spring mass system, m w^2 = k
29.03 w^2 = 116.67
w = 2 rad/s
at t= 0 , y = sqrt(3) ft = sqrt(3) x 0.3048 = 0.528 m
y = A sin(wt + phi)
at t = 0, v = 2ft/s = 2 x 0.3048 m /s = 0.61 m/s
total energy = mv^2 /2 + ky^2 /2 = kA^2 /2
(29.03 x 0.61^2 /2 ) + (116.67 x 0.528^2 / 2) = 116.67A^2 /2
A = 0.609 m
a) y = 0.609 sin(2t +phi)
at t-0, y = 0.528
0.528 = 0.609 sin(phi)
phi = pi/3
y = 0.609 m sin(2t + pi/3 )
b) A = 0.609 m
c) T = 2pi / w = 2pi/2 = pi Or 3.14 s
d) phase angle = pi/3 rad
e) y = 0
0 = 0.609 sin(2t + pi/3)
2t + pi/3 = pi
t = pi/3 s = 1.05 sec
f) t = 1.04 + 3.14 = 4.19 sec
g) 2t = 6pi - pi/3
t = 17pi/6 = 8.90 sec
h) 0.3048 m = 0.609 sin(2t + pi/3)
2t + pi/3 = sin^-1(0.3048/ 0.609) = pi/6
2t + pi/3 = (4n + 1) pi/6
t = (4n - 1)pi/12
n =0,1,2,3,....
i) t = (4n - 1)pi/12
n = 1,3,5,7, ....
j) t = (4n - 1)pi/12
n = 2,4,6 ....
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