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Question

For the damped oscillator system shown in above figure, the block has a mass of 1.50kg and the spring constant is 8.00N/m. The damping force is given by b(dx/dt), where b=230g/s. The block is pulled down 12.0cm and released. (a) Calculate the time required for the amplitude of the resulting oscillations to fall to one-third of its initial value. (b) How many oscillations are made by the block in this time?
1771918_259910bd1b374300990c6c26ece7ad6d.png

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Solution

(a) We want to solve ebt/2m=1/3 for t. We take the natural logarithm of both sides to obtain bt/2m=ln(1/3). Therefore, t=(2m/b)ln(1/3)=(2m/b)ln3. Thus,

t=2(1.50kg)0.230kg/sln3=14.3s.


(b) The angular frequency is

ω=kmb24m2= 8.0N/m1.50kg(0.230kg/s)24(1.50kg)2=2.31rad/s.

The period is T=2π/ω´=(2π)/(2.31rad/s)=2.72s and the number of oscillations is

t/T=(14.3s)/(2.72s)=5.27.

The displacement x(t) as a function of time is shown below. The amplitude, xmebt/2m, decreases exponentially with time.

1686307_1771918_ans_b68701868ba843f291a5782ffd2951df.png

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