If mass (M), length (L) and force (F) are considered as fundamental quantities then, find the dimensional formula for time

Options

A. `M^(1//2)L^(-1//2)F^(1//2)`
B. `M^(-1//2)L^(-1//2)F^(1//2)`
C. `M^(1//2)L^(-1//2)F^(1//2)`
D. `M^(1//2)L^(1//2)F^(-1//2)`

Answer

To find the dimensional formula for time (T) in terms of mass (M), length (L), and force (F), we can follow these steps:

### Step 1: Write down the dimensional formula for force
The dimensional formula for force (F) is given by:
\[ F = M \cdot a \]
where \( a \) (acceleration) has the dimensions \( L \cdot T^{-2} \). Therefore, the dimensional formula for force can be expressed as:
\[ [F] = [M][L][T^{-2}] = M^1 L^1 T^{-2} \]

### Step 2: Rearrange the equation to express time in terms of mass, length, and force
We can rearrange the equation for force to isolate time (T):
\[ [F] = M^1 L^1 T^{-2} \]
This can be rearranged as:
\[ T^{-2} = \frac{[F]}{[M][L]} \]
Taking the reciprocal gives:
\[ T^2 = \frac{[M][L]}{[F]} \]

### Step 3: Take the square root to find the dimensional formula for time
Now, taking the square root of both sides, we get:
\[ T = \sqrt{\frac{[M][L]}{[F]}} \]
This can be rewritten as:
\[ T = [M]^{1/2} [L]^{1/2} [F]^{-1/2} \]

### Step 4: Write the final dimensional formula for time
Thus, the dimensional formula for time (T) in terms of mass (M), length (L), and force (F) is:
\[ [T] = M^{1/2} L^{1/2} F^{-1/2} \]

### Conclusion
The dimensional formula for time in terms of mass, length, and force is:
\[ [T] = M^{1/2} L^{1/2} F^{-1/2} \]