`mx^(2) -bx +k = 0`
Find time after which of the energy will be come half of initial maximum value in damped forced oscillation .

Options

A. `t = m/b +1/2 ln2`
B. `t = m/b xx 2/3 ln2`
C. `t = m/2 - 1/2 ln2`
D. `t = m/b xx 1/2 ln 2`

Explanation

To solve the problem of finding the time after which the energy in a damped forced oscillation becomes half of its initial maximum value, we can follow these steps:

### Step 1: Understand the Energy in Damped Oscillation
In a damped oscillation, the energy \( E \) at any time \( t \) is proportional to the square of the displacement \( x(t) \):
\[ E(t) \propto x^2(t) \]
The maximum energy \( E_{\text{max}} \) occurs when the displacement is at its maximum amplitude \( A_0 \):
\[ E_{\text{max}} \propto A_0^2 \]

### Step 2: Write the Displacement Equation
For a damped oscillation, the displacement as a function of time can be expressed as:
\[ x(t) = A_0 e^{-\frac{b}{m} t} \]
where \( b \) is the damping coefficient and \( m \) is the mass.

### Step 3: Express Energy in Terms of Displacement
The energy at any time \( t \) can be written as:
\[ E(t) \propto x^2(t) = A_0^2 e^{-\frac{2b}{m} t} \]

### Step 4: Set Up the Equation for Half Energy
We want to find the time \( t \) when the energy becomes half of the maximum energy:
\[ E(t) = \frac{1}{2} E_{\text{max}} \]
Substituting the expressions for energy, we have:
\[ A_0^2 e^{-\frac{2b}{m} t} = \frac{1}{2} A_0^2 \]

### Step 5: Simplify the Equation
Dividing both sides by \( A_0^2 \) gives:
\[ e^{-\frac{2b}{m} t} = \frac{1}{2} \]

### Step 6: Take the Natural Logarithm
Taking the natural logarithm on both sides:
\[ -\frac{2b}{m} t = \ln\left(\frac{1}{2}\right) \]
Using the property of logarithms, we can express \( \ln\left(\frac{1}{2}\right) \) as:
\[ \ln\left(\frac{1}{2}\right) = -\ln(2) \]
Thus, we have:
\[ -\frac{2b}{m} t = -\ln(2) \]

### Step 7: Solve for Time \( t \)
Rearranging the equation gives:
\[ t = \frac{m \ln(2)}{2b} \]

### Final Result
Thus, the time after which the energy becomes half of its initial maximum value in damped forced oscillation is:
\[ t = \frac{m \ln(2)}{2b} \]