In the following equation, x, 1 and F represent respectively, displacement, time and force.
`F=a+bt+(1)/(c+d.x)+Asin(omegat+phi)`.
The dimensional formula for A.d is

Options

A. `[T^(-1)]`
B. `[L^(-1)]`
C. `[M^(-1)]`
D. `[TL^(-1)]`

Explanation

To find the dimensional formula for \( A \cdot d \) from the given equation:

\[ F = a + bt + \frac{1}{c + dx} + A \sin(\omega t + \phi) \]

we will follow these steps:

### Step 1: Understand the Dimensions of Each Term
The left-hand side of the equation represents force \( F \). The dimensional formula for force is given by:
\[ [F] = MLT^{-2} \]

### Step 2: Analyze the Right-Hand Side
Since the equation is an equality, all terms on the right-hand side must have the same dimensions as force \( F \).

### Step 3: Analyze the Term \( A \sin(\omega t + \phi) \)
The term \( \sin(\omega t + \phi) \) is dimensionless. Therefore, the dimensional formula of \( A \) must also be the same as that of force:
\[ [A] = MLT^{-2} \]

### Step 4: Analyze the Term \( \frac{1}{c + dx} \)
For the term \( \frac{1}{c + dx} \) to have the same dimensions as force, the dimensions of \( c + dx \) must also be \( MLT^{-2} \).

### Step 5: Determine the Dimensions of \( dx \)
The term \( dx \) has dimensions:
\[ [dx] = [d][x] \]
where \( [x] = L \) (displacement).

### Step 6: Find the Dimensions of \( c \)
Since \( c + dx \) must have the same dimensions as force, we can express \( [c] \):
\[ [c + dx] = [F] = MLT^{-2} \]
This implies that \( [c] \) must also have dimensions of \( MLT^{-2} \) because \( dx \) must also contribute to the dimensions of force.

### Step 7: Determine the Dimensions of \( d \)
From \( [dx] = [d][x] \):
\[ [d][L] = MLT^{-2} \]
Thus, we can express \( [d] \):
\[ [d] = \frac{MLT^{-2}}{L} = MLT^{-2} \cdot L^{-1} = MLT^{-2}L^{-1} = MLT^{-2}L^{-1} \]
So, we have:
\[ [d] = MLT^{-2}L^{-1} = ML^{-1}T^{-2} \]

### Step 8: Calculate the Dimensions of \( A \cdot d \)
Now we can find the dimensions of \( A \cdot d \):
\[ [A \cdot d] = [A][d] = (MLT^{-2})(ML^{-1}T^{-2}) = M^2L^{0}T^{-4} \]

### Step 9: Final Result
The dimensional formula for \( A \cdot d \) simplifies to:
\[ [A \cdot d] = M^2T^{-4} \]

### Conclusion
The dimensional formula for \( A \cdot d \) is:
\[ \text{Dimensional Formula for } A \cdot d = M^2T^{-4} \]