In the above problem, the maximum positive displacement x is

Options

A. `2 sqrt(3) m`
B. 2 m
C. 4 m
D. `sqrt(2)m`

Answer

To solve the problem of finding the maximum positive displacement \( x \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find the maximum positive displacement \( x \) where the kinetic energy \( K \) of a particle is given by the equation:
\[ K = 4x - \frac{x^3}{3} \]

2. **Setting Kinetic Energy to Zero**: At maximum displacement, the velocity of the particle is zero. Therefore, the kinetic energy \( K \) at this point is also zero:
\[ K = 0 \]

3. **Setting Up the Equation**: We can set the kinetic energy equation to zero:
\[ 4x - \frac{x^3}{3} = 0 \]

4. **Rearranging the Equation**: To solve for \( x \), we rearrange the equation:
\[ 4x = \frac{x^3}{3} \]

5. **Multiplying Through by 3**: To eliminate the fraction, multiply both sides by 3:
\[ 12x = x^3 \]

6. **Rearranging to Form a Polynomial**: Rearranging gives us:
\[ x^3 - 12x = 0 \]

7. **Factoring the Equation**: We can factor out \( x \):
\[ x(x^2 - 12) = 0 \]

8. **Finding the Roots**: This gives us two factors:
- \( x = 0 \) (which is not the maximum positive displacement)
- \( x^2 - 12 = 0 \)

9. **Solving for \( x \)**: Solving \( x^2 - 12 = 0 \) gives:
\[ x^2 = 12 \implies x = \sqrt{12} = 2\sqrt{3} \]

10. **Conclusion**: Therefore, the maximum positive displacement \( x \) is:
\[ x = 2\sqrt{3} \, \text{m} \]

### Final Answer:
The maximum positive displacement \( x \) is \( 2\sqrt{3} \, \text{m} \).