A mass weighing 64 pounds stretches a spring 0.32 foot. The mass is initially re

Question

A mass weighing 64 pounds stretches a spring 0.32 foot. The mass is initially released from a point 8 inches above the equilibrium position with a downward velocity of 5 ft/s. (a) Find the equation of motion. (b) What are the amplitude and period of motion? (c) How many complete cycles will the mass have completed at the end of 3 pi seconds? (d) At what time does the mass pass through the equilibrium position heading downward for the second time? (e) At what times does the mass attain its extreme displacements on either side of the equilibrium position? (f) What is the position of the mass at t = 3 s? (g) What is the instantaneous velocity at t = 3 s? (h) What is the acceleration at t = 3 s? (i) What is the instantaneous velocity at the times when the mass passes through the equilibrium position? (j) At what times is the mass 5 inches below the equilibrium position? (k) At what times is the mass 5 inches below the equilibrium position heading in the upward direction

Explanation / Answer

m = 64 pounds = 0.454 x 64 kg = 29 kg

d = 0.32 foot = 0.0975 m

at equilibrium position, Fnet = kx - m g = 0

k (0.0975) = 29 x 9.8

k = 2914.87 N/m

w = sqrt(k/m) = 10 rad/s

A = 8 inches

(A) y = A cos(wt)

y = 8 inch cos(10 t)

(B) A = 8 inch

T = 2pi/w = 2 pi / 10 = 0.2 pi sec OR 0.628 sec

(C) n = t / T = 3 pi / (0.2 pi) = 15 oscillations

(D) 0= 8 inch cos(10t)

10 t = (2n + 1) pi / 2

and for heading donward,

10t = (4n + 1) pi/2

for first time, n = 0

for second time, n = 1

10 t = 5 pi / 2

t = 0.25 pi sce

(E) for above,

cos(10t) = 1

10t = n 2pi

n = 0 , 1 , 2, ....

t = 0.2 n pi (for above extreme positon)

where n = 0,1,2,3,.....

for below,

cos(10t) = -1

10t = (2n + 1) pi

t = (2n + 1) pi / 10

where n = 0, 1,2,3......

(F) t = 3 sec

y = 1.23 inches

(G) v = dy/dt = - 80 sin(10t) inch/s

at t = 3 sec

v = 79.04 inch/s

(h) a =dv/dt = - 800 cos(10t) inch/s^2

a = - 123.4 inch/s^2

(i) 80 inch/s

(j) -5 = 8 cos(10t)

cos(10t) = -0.625

10t = 0.715pi + n 2pi

t = (2n + 0.715) pi/10

(k) 10t = 1.285pi + n 2pi

t = (2n + 1.285) pi / 10

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