Given information:
- The amplitude of a pendulum executing damped simple harmonic motion falls to 1/3 the original value after 100 oscillations.
- The amplitude falls to S times the original value after 200 oscillations.

To find: The value of S.

Solution:

Let A0 be the original amplitude of the pendulum.

After 100 oscillations, the amplitude falls to 1/3 of the original value:
Amplitude after 100 oscillations = A0/3

After 200 oscillations, the amplitude falls to S times the original value:
Amplitude after 200 oscillations = S * A0

We can use the concept of damping factor (λ) to relate the amplitude after a certain number of oscillations to the original amplitude.

Damping factor:
The damping factor (λ) is defined as the ratio of the amplitude after one complete oscillation (A1) to the amplitude before that oscillation (A0).

Using the formula for damped simple harmonic motion, we can write:
A1 = A0 * e^(-λT)

Where A1 is the amplitude after one complete oscillation, T is the time period of the pendulum, and e is the base of the natural logarithm.

Calculating damping factor:

We are given that the amplitude falls to 1/3 of the original value after 100 oscillations. Plugging in the values, we get:
1/3 = A0 * e^(-λT1)

Similarly, after 200 oscillations, the amplitude falls to S times the original value:
S = A0 * e^(-λT2)

Taking the ratio of the above equations, we get:
(S/1/3) = e^(-λT2) / e^(-λT1)

Simplifying the equation, we get:
S/1/3 = e^(-λT2 + λT1)
S/1/3 = e^(-λ(T2 - T1))

Since T2 - T1 is the time period between 100 and 200 oscillations, it is equal to the time period of 100 oscillations.

S/1/3 = e^(-λT1)

We are given that S is the ratio of the amplitude after 200 oscillations to the original amplitude. So, S is equal to e^(-λT1).

Since S is the value of e^(-λT1), and T1 is the time period of 100 oscillations, we can conclude that S is equal to e^(-λ100).

Conclusion:

From the above calculations, we can see that S is equal to e^(-λ100), which means that S is a positive value less than 1. The only option that satisfies this condition is option A) 1/9.

Hence, the correct answer is option A) 1/9.

Correct answer is option.A