A lightly damped harmonic oscillator, with a damping force proportional to its speed, is oscillating with an amplitude of 0.500 cm at time t = 0. When t = 8.20 s, the amplitude has died down to 0.400 cm. At what value of t will the oscillations have an amplitude of 0.250 cm?

Answer:

since the damping force is prop to speed, the amplitude decreases exponentially, i.e. we can write

  amplitude = initial amplitude * e^-Kt    where K is a constant

You can now write this for both of the given situations:

    first:   0.400 = 0.500 e^-K*8.20

    second:   0.250 = 0.500 e^-K*t

Notice that you can solve for K in the first equation:

    0.400 / 0.500 = e^-K*8.20    

take ln of both sides

   ln0.8 = -8.20K

K = - ln0.8 / 8.20 = 0.027213

Now use this in the second equation, and solve for time

0.250 = 0.500 e^-0.027213*t

ln(0.250/0.500) = -0.027213*t

   t = 25.5 seconds is the time at which the amplitude will reach 0.250