A transverse wave is tranvelling in a string at any moment a small element 'dx' is at inclination `30^@` with the direction of propagation of the wave. After some time interval its inclination changes to `60^@` with direction of propagation. Potential energy of this small element is initially `U_0` and finally it is `KU_0`, value of K is

Options

A. 2
B. 9
C. 3
D. 4

Answer

To solve the problem, we need to analyze the relationship between the potential energy of a small element of the string and its angle of inclination with the direction of wave propagation.

### Step-by-Step Solution:

1. **Understanding Potential Energy in Transverse Waves**:
The potential energy (U) of a small element of the string in a transverse wave is directly proportional to the square of the sine of the angle (θ) that the element makes with the direction of wave propagation. This can be expressed mathematically as:
\[ U \propto \sin^2(\theta) \]

2. **Initial Condition**:
Initially, the angle of inclination is \(30^\circ\). Therefore, the initial potential energy \(U_0\) can be expressed as:
\[ U_0 = k_1 \sin^2(30^\circ) \]
where \(k_1\) is a proportionality constant. We know that:
\[ \sin(30^\circ) = \frac{1}{2} \]
Thus,
\[ U_0 = k_1 \left(\frac{1}{2}\right)^2 = k_1 \cdot \frac{1}{4} \]

3. **Final Condition**:
After some time, the angle changes to \(60^\circ\). The final potential energy \(KU_0\) can be expressed as:
\[ KU_0 = k_2 \sin^2(60^\circ) \]
where \(k_2\) is another proportionality constant. We know that:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]
Thus,
\[ KU_0 = k_2 \left(\frac{\sqrt{3}}{2}\right)^2 = k_2 \cdot \frac{3}{4} \]

4. **Relating Initial and Final Potential Energies**:
Since we have \(U_0\) and \(KU_0\), we can set up the following relationship:
\[ KU_0 = \frac{3}{4} k_2 \]
and
\[ U_0 = \frac{1}{4} k_1 \]

5. **Finding the Value of K**:
We can express the ratio of the two potential energies:
\[ \frac{U_0}{KU_0} = \frac{\frac{1}{4} k_1}{\frac{3}{4} k_2} \]
Simplifying this gives:
\[ \frac{1}{K} = \frac{1}{3} \]
Therefore, we can find \(K\):
\[ K = 3 \]

### Final Answer:
The value of \(K\) is \(3\).

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