If the displacement y of a particle is y = A sin (pt + qx), then the dimension of 'p.q' is -

Options

A. the dimensions of p are `M^(@)L^(@)T`
B. the dimensions of q are `M^(@)LT^(@)`
C. the dimensions of p.q are `M^(@)LT^(-1)`
D. the dimensions of pq are `M^(@)LT^(-1)`

Answer

To find the dimension of \( p \cdot q \) from the given displacement equation \( y = A \sin(pt + qx) \), we can follow these steps:

### Step 1: Understand the Equation
The displacement \( y \) is given by the equation \( y = A \sin(pt + qx) \). Here, \( A \) is a constant amplitude, and \( pt + qx \) is the argument of the sine function.

### Step 2: Identify the Argument of the Sine Function
For the sine function to be defined, the argument \( pt + qx \) must be dimensionless. This means that the dimensions of \( pt \) and \( qx \) must both equal 1 (dimensionless).

### Step 3: Analyze \( pt \)
Let’s analyze \( pt \):
- The dimension of time \( t \) is \( [T] \).
- Therefore, for \( pt \) to be dimensionless, we have:
\[ [p] \cdot [T] = 1 \]
This implies:
\[ [p] = [T]^{-1} \]

### Step 4: Analyze \( qx \)
Now, let’s analyze \( qx \):
- The dimension of position \( x \) is \( [L] \).
- Therefore, for \( qx \) to be dimensionless, we have:
\[ [q] \cdot [L] = 1 \]
This implies:
\[ [q] = [L]^{-1} \]

### Step 5: Find the Product \( p \cdot q \)
Now, we can find the dimensions of the product \( p \cdot q \):
\[ [p \cdot q] = [p] \cdot [q] = [T]^{-1} \cdot [L]^{-1} \]
This results in:
\[ [p \cdot q] = [L]^{-1} [T]^{-1} \]

### Step 6: Conclusion
Thus, the dimension of \( p \cdot q \) is:
\[ [L]^{-1} [T]^{-1} \]

### Summary
The final answer is that the dimension of \( p \cdot q \) is \( [L]^{-1} [T]^{-1} \).

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