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. (a)0.25
B. (b)0.5
C. (c)2
D. (d)0.125

Explanation

To solve the problem step by step, we will use the formula for damped harmonic motion, which describes how the amplitude decreases over time due to damping forces such as air friction.

### Step 1: Understand the given information
- The amplitude 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 (total of 12 seconds).

### Step 2: Use the formula for damped harmonic motion
The amplitude \( A(t) \) at time \( t \) can be expressed as:
\[ A(t) = A_0 e^{-kt} \]
where:
- \( A_0 \) is the initial amplitude,
- \( k \) is the damping constant,
- \( t \) is the time in seconds.

### Step 3: Set up the equation for the first 4 seconds
From the problem, we know that after 4 seconds:
\[ A(4) = 0.5 A_0 \]
Substituting into the formula:
\[ 0.5 A_0 = A_0 e^{-4k} \]

### Step 4: Simplify the equation
Dividing both sides by \( A_0 \) (assuming \( A_0 \neq 0 \)):
\[ 0.5 = e^{-4k} \]

### Step 5: Take the natural logarithm of both sides
Taking the natural logarithm:
\[ \ln(0.5) = -4k \]
Thus,
\[ k = -\frac{\ln(0.5)}{4} \]

### Step 6: Calculate the amplitude after another 8 seconds (total 12 seconds)
Now we need to find the amplitude at \( t = 12 \) seconds:
\[ A(12) = A_0 e^{-12k} \]

### Step 7: Substitute the value of \( k \)
Substituting the expression for \( k \):
\[ A(12) = A_0 e^{-12 \left(-\frac{\ln(0.5)}{4}\right)} = A_0 e^{3 \ln(0.5)} \]

### Step 8: Simplify using properties of exponents
Using the property \( e^{\ln(x)} = x \):
\[ A(12) = A_0 (0.5)^3 \]
Calculating \( (0.5)^3 \):
\[ A(12) = A_0 \cdot 0.125 \]

### Step 9: Conclusion
Thus, the amplitude after 12 seconds is:
\[ A(12) = 0.125 A_0 \]
This means the amplitude becomes 0.125 times the original magnitude.

### Final Answer
The amplitude becomes \( 0.125 \) times its original magnitude in another 8 seconds.

---