Given that the displacement of a particle `x=A^(2)sin^(2)(Kt)`, where 't' denotes the time. The dimension of K is same as that of :

Options

A. (a)hertz
B. (b)metre
C. (c)radian
D. (d)second

Answer

To solve the problem, we need to analyze the given displacement equation and determine the dimensions of the constant \( K \).

### Step-by-Step Solution:

1. **Identify the given equation**:
The displacement of the particle is given by:
\[ x = A^2 \sin^2(Kt) \]
where \( x \) is the displacement, \( A \) is a constant, \( K \) is a constant, and \( t \) is the time.

2. **Understand the sine function**:
The sine function, \( \sin(\theta) \), is dimensionless. This means that the argument of the sine function must also be dimensionless.

3. **Set up the condition for dimensional analysis**:
Since \( Kt \) is the argument of the sine function, it must be dimensionless:
\[ [Kt] = 1 \quad \text{(dimensionless)} \]
This implies that the dimensions of \( K \) multiplied by the dimensions of \( t \) must equal 1.

4. **Determine the dimensions of time**:
The dimension of time \( t \) is given by:
\[ [t] = T \]
where \( T \) represents the dimension of time.

5. **Express the dimension of \( K \)**:
From the equation \( [Kt] = 1 \), we can express the dimension of \( K \):
\[ [K] \cdot [t] = 1 \implies [K] \cdot T = 1 \implies [K] = T^{-1} \]

6. **Identify the physical quantity corresponding to \( T^{-1} \)**:
The dimension \( T^{-1} \) corresponds to frequency. The SI unit of frequency is Hertz (Hz).

7. **Conclusion**:
Therefore, the dimension of \( K \) is the same as that of frequency, which is Hertz.

### Final Answer:
The dimension of \( K \) is the same as that of **Hertz**.

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