If initial amplitude during a damped oscillation of mass m is 12 cm and after 2s it reduces to 6cm then find damping constant(b)

Options

A. `m ln(2)`
B. `2m`
C. `m^2 ln2`
D. `(2m)/(m^2)`

Answer

To solve the problem of finding the damping constant \( b \) in a damped oscillation, we can follow these steps:

### Step 1: Understand the Damped Oscillation Formula
The amplitude of a damped oscillation decreases over time according to the formula:
\[ A(t) = A_0 e^{-\frac{bt}{2m}} \]
where:
- \( A(t) \) is the amplitude at time \( t \),
- \( A_0 \) is the initial amplitude,
- \( b \) is the damping constant,
- \( m \) is the mass,
- \( t \) is the time.

### Step 2: Substitute the Given Values
From the problem, we know:
- Initial amplitude \( A_0 = 12 \) cm,
- Amplitude after \( t = 2 \) s is \( A(2) = 6 \) cm.

Substituting these values into the formula gives:
\[ 6 = 12 e^{-\frac{b \cdot 2}{2m}} \]

### Step 3: Simplify the Equation
We can simplify the equation:
\[ \frac{6}{12} = e^{-\frac{b \cdot 2}{2m}} \]
This simplifies to:
\[ \frac{1}{2} = e^{-\frac{b \cdot 2}{2m}} \]

### Step 4: Take the Natural Logarithm
Taking the natural logarithm of both sides:
\[ \ln\left(\frac{1}{2}\right) = -\frac{b \cdot 2}{2m} \]
This can be rewritten as:
\[ -\ln(2) = -\frac{b}{m} \]

### Step 5: Solve for the Damping Constant \( b \)
Rearranging the equation gives:
\[ b = m \ln(2) \]

### Conclusion
Thus, the damping constant \( b \) is given by:
\[ b = m \ln(2) \]