Amplitude of a swing decreases to 0.5 times its original magnitude in 4s due to damping by air friction. Its amplitude becomes how many times of the original magnitude in another 8s?

Options

A. 0.25
B. 0.5
C. 2
D. 0.125

Explanation

To solve the problem, we need to analyze the damped harmonic motion of the swing. The amplitude decreases due to damping, and we can use the formula for damped harmonic motion to find the amplitude after a certain time.

### Step-by-Step Solution:

1. **Understanding the Problem**:
The amplitude of the swing decreases to 0.5 times its original value in 4 seconds. We need to find out how many times the original amplitude it becomes after another 8 seconds (totaling 12 seconds).

2. **Using the Damped Harmonic Motion Formula**:
The general formula for the amplitude \( A \) at time \( t \) in damped harmonic motion is given by:
\[ A(t) = A_0 e^{-kt} \]
where \( A_0 \) is the initial amplitude, \( k \) is the damping constant, and \( t \) is the time.

3. **Finding the Damping Constant \( k \)**:
From the problem, we know that after 4 seconds, the amplitude becomes 0.5 times the original amplitude:
\[ A(4) = 0.5 A_0 \]
Substituting into the formula:
\[ 0.5 A_0 = A_0 e^{-4k} \]
Dividing both sides by \( A_0 \) (assuming \( A_0 \neq 0 \)):
\[ 0.5 = e^{-4k} \]
Taking the natural logarithm of both sides:
\[ \ln(0.5) = -4k \]
Therefore, we can express \( k \) as:
\[ k = -\frac{\ln(0.5)}{4} \]

4. **Calculating the Amplitude After 12 Seconds**:
Now we need to find the amplitude after a total of 12 seconds:
\[ A(12) = A_0 e^{-k \cdot 12} \]
Substituting the value of \( k \):
\[ A(12) = A_0 e^{-12 \left(-\frac{\ln(0.5)}{4}\right)} \]
Simplifying this gives:
\[ A(12) = A_0 e^{3 \ln(0.5)} \]
Using the property of logarithms \( e^{\ln(x^n)} = x^n \):
\[ A(12) = A_0 (0.5)^3 \]
Calculating \( (0.5)^3 \):
\[ A(12) = A_0 \cdot 0.125 \]

5. **Final Result**:
Thus, the amplitude after another 8 seconds (totaling 12 seconds) is:
\[ A(12) = 0.125 A_0 \]

### Conclusion:
The amplitude becomes 0.125 times the original magnitude after another 8 seconds.