The graph shows the displacement from equilibrium of a mass-spring system as a f

Question

The graph shows the displacement from equilibrium of a mass-spring system as a function of time after the vertically hanging system was set in motion at time t = 0. Assume that the units of time are seconds, and the units of displacement are centimeters. The first t-intercept is (0.25, 0) and the first minimum has coordinates (0.75, -4). (a) What is the period T of the periodic motion? T = ______ seconds (b) What is the frequency f n Hertz? What is the angular frequency omega in radians/second? F = ______ Hertz omega = ______ radians/second (d) Determine the amplitude A and the phase angle gamma (in radians), and express the displacement in the form y(t) = A cos (omega t - gamma), with y in meters. y(t) = ______ meters (e) With what initial displacement y(0) and initial velocity y'(0) was the system set into motion? y(0) = _____ meters y'(0) = meters/second

Explanation / Answer

a)Time Period=(0.75-0.25)*4

T=2 second

b) f=1/T

f=1/2=0.5 Hertz

=2f=2*0.5==3.14 rad/sec.

d) y(t)=Acos(t-)

given y(0.25)=0

0=Acos(/4-)

=-/4

Given y(0.75)=-4

-4=Acos(3/4+/4)

A=4

y(t)=4cos(t+/4)

e) y'(t)=-4sin(t+/4)

y'(0)=-4/1.414=-8.88 m/s

y(0)=4cos(/4)=2.82 meter

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