In the relation `y=acos(omegat-kx)`, the dimensional formula for `k` is

Options

A. `M^(0)L^(0)T^(-1)`
B. `M^(0)L^(0)T^(0)`
C. `M^(0)L^(-1)T^(0)`
D. `MLT^(-2)`

Answer

To find the dimensional formula for \( k \) in the relation \( y = a \cos(\omega t - kx) \), we can follow these steps:

### Step 1: Understand the equation
The equation \( y = a \cos(\omega t - kx) \) represents a wave, where:
- \( y \) is the displacement,
- \( a \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number,
- \( t \) is time,
- \( x \) is position.

### Step 2: Identify the dimensions of the terms inside the cosine
The argument of the cosine function, \( \omega t - kx \), must be dimensionless. This means that the dimensions of \( \omega t \) and \( kx \) must be the same.

### Step 3: Determine the dimensions of \( \omega t \)
- The angular frequency \( \omega \) has dimensions of \( [T^{-1}] \) (inverse of time).
- Time \( t \) has dimensions of \( [T] \).

Thus, the dimensions of \( \omega t \) are:
\[ [\omega t] = [T^{-1}] \cdot [T] = [1] \quad (\text{dimensionless}) \]

### Step 4: Determine the dimensions of \( kx \)
- The wave number \( k \) has dimensions we need to find.
- Position \( x \) has dimensions of \( [L] \) (length).

Thus, the dimensions of \( kx \) are:
\[ [kx] = [k] \cdot [L] \]

### Step 5: Set the dimensions equal
Since both \( \omega t \) and \( kx \) must be dimensionless, we can equate their dimensions:
\[ [\omega t] = [kx] \]
This gives us:
\[ [1] = [k] \cdot [L] \]

### Step 6: Solve for the dimensions of \( k \)
From the equation \( [1] = [k] \cdot [L] \), we can express \( k \) as:
\[ [k] = [L^{-1}] \]

### Step 7: Write the final dimensional formula
The dimensional formula for \( k \) can be expressed as:
\[ [k] = [M^0 L^{-1} T^0] \]

### Conclusion
Thus, the dimensional formula for \( k \) is:
\[ \boxed{M^0 L^{-1} T^0} \]