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. hertz
B. metre
C. radian
D. second

Answer

To find the dimension of \( K \) in the equation \( x = A^2 \sin^2(Kt) \), we need to ensure that the argument of the sine function, \( Kt \), is dimensionless. Let's break this down step by step.

### Step 1: Understand the Argument of the Sine Function
The sine function takes an angle as its argument, and angles are dimensionless quantities. Therefore, for \( Kt \) to be dimensionless, the product must not have any units.

### Step 2: Analyze the Dimensions
Let’s denote the dimensions of \( x \) (displacement) and \( A \) (a constant) as follows:
- The dimension of displacement \( x \) is \( [L] \) (length).
- The dimension of \( A^2 \) will also be \( [L] \) since it is a constant related to displacement.

### Step 3: Set Up the Equation
Since \( Kt \) must be dimensionless, we can express this as:
\[ [K] \cdot [T] = 1 \]
where \( [T] \) is the dimension of time.

### Step 4: Solve for the Dimension of \( K \)
Rearranging the equation gives us:
\[ [K] = \frac{1}{[T]} \]
This means that the dimension of \( K \) is \( [T]^{-1} \).

### Step 5: Identify the Unit of \( K \)
The dimension \( [T]^{-1} \) corresponds to frequency, which is measured in Hertz (Hz).

### Conclusion
Thus, the dimension of \( K \) is \( [T]^{-1} \), which is equivalent to the unit of frequency.

### Final Answer
The dimension of \( K \) is \( [T]^{-1} \), or Hertz (Hz).

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