1. A mass is suspended by a spring attached to the ceiling. Initially, the mass
ID: 2920019 • Letter: 1
Question
1. A mass is suspended by a spring attached to the ceiling. Initially, the mass rests at its equilibrium point. The mass is pulled down 10.0 cm, held steady, and released. A high speed camera is used to observe that after 12.73 seconds, at the beginning of its seventh full oscillation, the mass is displaced by 6.83 centimeters.
(a) Write a function of the form d(t) = kect sin(t) or d(t) = kect cos(t)
that models the displacement of the mass, in meters, above its equilibrium point t seconds after it was initially released. Round values of all constants to three significant figures.
(b) How long will it take for the oscillations to be damped so that their amplitude is 1% of the original amplitude? Round your answer to three significant figures
Explanation / Answer
1)
d(t) = kect cos(t)
function models the displacement of the mass, in meters, above its equilibrium point
initially mass is pulled down 10.0 cm , held steady, and released.
so k = -10
period of 6 oscillations =12.73 seconds
=>periof of one oscillation =12.73/6
=>2/ =12.73/6
=> =12/12.73
d(12.73) =6.83
=> -10ec*12.73 cos((12/12.73)*12.73) =-6.83
=> 10ec*12.73 cos(12) =6.83
=> ec*12.73 *1 =6.83/10
=> ec*12.73 =0.683
=>-12.73c =ln(0.683)
=>c =-(ln(0.683))/12.73
=>c =0.02994975800560463383119632484234
=>c =0.0299
model is d(t)=-10e0.0299t cos((12/12.73)t)
(b)
|d(t)|= (1/100)*10.0
=>|-10e0.0299t cos((12/12.73)t)|= 0.1
=>t = 151.578, 152.839
it takes 152 seconds so that their amplitude is 1% of the original amplitude
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