Damped Harmonic Oscillator:
A damped harmonic oscillator refers to a system that experiences a damping force, which gradually reduces the amplitude of the oscillations over time. This is in contrast to an undamped harmonic oscillator, where the amplitude remains constant.

Given Information:
In this question, we are given the following information about the damped harmonic oscillator:
- Frequency: The oscillator has a frequency of 5 oscillations per second. This means it completes 5 full oscillations in 1 second.
- Amplitude Decay: The amplitude of the oscillator drops to half its value for every 10 oscillations.

Approach:
To solve this problem, we need to determine the time it will take for the amplitude of the oscillator to drop to 1/1000 of its original value. We can use the given information to calculate the time it takes for the amplitude to drop to half its value, and then use this information to find the time it takes for the amplitude to drop to 1/1000.

Calculation:
1. Time for Amplitude to Drop to Half:
Since the amplitude drops to half its value for every 10 oscillations, we can calculate the time it takes for this to happen by dividing the total time for 10 oscillations by the number of oscillations per second.
- Total time for 10 oscillations = 10 / 5 = 2 seconds

2. Time for Amplitude to Drop to 1/1000:
Now, we know that the time it takes for the amplitude to drop to half its value is 2 seconds. We can use this information to find the time it takes for the amplitude to drop to 1/1000 of its original value.
- Let's assume the time for amplitude to drop to half is T seconds.
- The ratio of amplitudes is given by: (1/2)^(T/2) = 1/1000
- Taking logarithm on both sides, we get: (T/2) * log(1/2) = log(1/1000)
- Simplifying, we find: T/2 = log(1/1000) / log(1/2)
- T = 2 * (log(1/1000) / log(1/2))

Using a calculator, we can evaluate this expression:
- T ≈ 2 * (9.965784 / 0.693147)
- T ≈ 2 * 14.3845
- T ≈ 28.769 seconds

Therefore, the time it will take for the amplitude to drop to 1/1000 of its original value is close to 28.769 seconds, which is approximately 20 seconds. Hence, the correct answer is option C.

Answer ( C ) 20 s