Equation of motion for a particle performing damped harmonic oscillation is given as `x=e^(-1t) cos(10pit+phi)`. The time when amplitude will half of the initial is :

Options

A. 27s
B. 4s
C. 13s
D. 7s

Explanation

To solve the problem step by step, we will analyze the given equation of motion for a particle performing damped harmonic oscillation, which is expressed as:

\[ x = e^{-t} \cos(10\pi t + \phi) \]

### Step 1: Identify the Amplitude
The amplitude \( A \) of the damped harmonic motion can be expressed as:

\[ A(t) = A_0 e^{-kt} \]

where \( A_0 \) is the initial amplitude and \( k \) is the damping constant. From the given equation, we can see that:

\[ A(t) = e^{-t} \]

This indicates that \( k = 1 \).

### Step 2: Set Up the Equation for Half Amplitude
We need to find the time \( t \) when the amplitude is half of the initial amplitude. Therefore, we set up the equation:

\[ A(t) = \frac{A_0}{2} \]

Substituting the expression for \( A(t) \):

\[ e^{-t} = \frac{A_0}{2} \]

### Step 3: Solve for \( t \)
Since we are looking for the time when the amplitude is half, we can express the initial amplitude \( A_0 \) as \( A_0 = e^{0} = 1 \) for simplicity. Thus, we rewrite the equation as:

\[ e^{-t} = \frac{1}{2} \]

To solve for \( t \), we take the natural logarithm of both sides:

\[ -t = \ln\left(\frac{1}{2}\right) \]

This simplifies to:

\[ t = -\ln\left(\frac{1}{2}\right) \]

Using the property of logarithms, we can rewrite this as:

\[ t = \ln(2) \]

### Step 4: Calculate the Value of \( t \)
The natural logarithm of 2 is approximately:

\[ \ln(2) \approx 0.693 \]

Thus, the time when the amplitude will be half of the initial amplitude is:

\[ t \approx 0.693 \text{ seconds} \]

### Final Answer
The time when the amplitude will be half of the initial amplitude is approximately:

\[ t \approx 0.693 \text{ seconds} \]

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