The amplitude of a S.H.O. reduces to 1/3 in first 20 secs. then in first 40 sec. its amplitude becomes -

Options

A. `1/3`
B. `1/9`
C. `1/27`
D. `1/sqrt3`

Explanation

To solve the problem step by step, we will use the formula for the amplitude of damped oscillations in simple harmonic motion (SHM). The amplitude of a damped oscillator at time \( t \) is given by:

\[ x(t) = x_0 e^{-\lambda t} \]

where:
- \( x(t) \) is the amplitude at time \( t \),
- \( x_0 \) is the initial amplitude,
- \( \lambda \) is the damping constant,
- \( t \) is the time.

### Step 1: Determine the amplitude after 20 seconds

According to the problem, the amplitude reduces to \( \frac{1}{3} \) of its original value after 20 seconds. This can be expressed as:

\[ x(20) = \frac{x_0}{3} \]

Using the formula for damped oscillations, we can set up the equation:

\[ \frac{x_0}{3} = x_0 e^{-\lambda \cdot 20} \]

### Step 2: Simplify the equation

We can cancel \( x_0 \) from both sides (assuming \( x_0 \neq 0 \)):

\[ \frac{1}{3} = e^{-\lambda \cdot 20} \]

### Step 3: Take the natural logarithm

Taking the natural logarithm of both sides gives:

\[ \ln\left(\frac{1}{3}\right) = -\lambda \cdot 20 \]

### Step 4: Solve for \( \lambda \)

Rearranging the equation to solve for \( \lambda \):

\[ \lambda = -\frac{\ln\left(\frac{1}{3}\right)}{20} \]

### Step 5: Determine the amplitude after 40 seconds

Now we need to find the amplitude after 40 seconds. We use the same formula:

\[ x(40) = x_0 e^{-\lambda \cdot 40} \]

Substituting \( \lambda \):

\[ x(40) = x_0 e^{-40 \cdot \left(-\frac{\ln\left(\frac{1}{3}\right)}{20}\right)} \]

### Step 6: Simplify the expression

This simplifies to:

\[ x(40) = x_0 e^{2 \ln\left(\frac{1}{3}\right)} \]

Using the property of logarithms \( e^{\ln(a^b)} = a^b \):

\[ x(40) = x_0 \left(\frac{1}{3}\right)^2 \]

### Step 7: Final result

Thus, the amplitude after 40 seconds is:

\[ x(40) = \frac{x_0}{9} \]

### Conclusion

The amplitude of the simple harmonic oscillator after 40 seconds is \( \frac{x_0}{9} \).

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