A damped oscillator is released from rest with an initial displacement of 10.00 cm. At the end of the first complete oscillation the displacement reaches 9.05 cm. When 4 more oscillations are completed, what is the displacement reached?
A) 7.41 cm
B) 6.71 cm
C) 6.07 cm
D) 5.49 cm
E) 5.25 cm

Question

Quizplus · Accepted Answer

We can use the equation for the amplitude of a damped oscillator:
A = A0 e^(-βt)
where A is the amplitude at time t, A0 is the initial amplitude, β is the damping coefficient.

From the statement of the problem: 
- A0 = 10.00 cm, 
- A = 9.05 cm after the first complete oscillation (T is the period of oscillation): 
9.05 = 10.00 e^(-βT), 
- we want to find A after 5 more oscillations: 
A = A0 e^(-5βT).

We can solve for βT by taking the natural logarithm of both sides of the equation for the first complete oscillation: 
ln(9.05) = ln(10.00) - βT, 
βT = ln(10.00) - ln(9.05) ≈ 0.0996 s^-1.

Substituting this into the equation for A after 5 more oscillations: 
A = 10.00 e^(-5βT) = 10.00 e^(-0.498 s^-1) ≈ 6.07 cm.

Therefore, the answer is C.

A Damped Oscillator Is Released from Rest with an Initial