If units of length, mass and force are chosen as fundamental units, the dimensions of time would be :

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 L^(-1//2) F^(-1//2)`

Explanation

To find the dimensions of time when length, mass, and force are chosen as fundamental units, we can follow these steps:

### Step 1: Understand the relationship between force, mass, length, and time.
The fundamental relationship we need to recall is Newton's second law of motion, which states that force (F) is equal to mass (m) multiplied by acceleration (a). Mathematically, this can be expressed as:
\[ F = m \cdot a \]
Acceleration (a) can be defined as the change in velocity (v) over time (t), and velocity is defined as the change in displacement (length, L) over time (t). Therefore, we can express acceleration as:
\[ a = \frac{L}{T^2} \]

### Step 2: Substitute the expression for acceleration into the force equation.
Now, substituting the expression for acceleration into the force equation gives us:
\[ F = m \cdot \frac{L}{T^2} \]

### Step 3: Rearrange the equation to isolate time (T).
Rearranging the equation to solve for time (T), we have:
\[ F = \frac{mL}{T^2} \]
Multiplying both sides by \(T^2\) gives:
\[ F \cdot T^2 = mL \]
Now, dividing both sides by F gives:
\[ T^2 = \frac{mL}{F} \]
Taking the square root of both sides, we find:
\[ T = \sqrt{\frac{mL}{F}} \]

### Step 4: Express the dimensions in terms of fundamental units.
Now we need to express the dimensions of time (T) in terms of mass (M), length (L), and force (F). We know that the dimensions of force (F) can be expressed as:
\[ [F] = [M][L][T^{-2}] \]
Thus, we can express the dimensions of time as:
\[ [T] = \sqrt{\frac{[M][L]}{[F]}} \]

### Step 5: Substitute the dimensions of force into the equation.
Substituting the dimensions of force into our equation gives:
\[ [T] = \sqrt{\frac{[M][L]}{[M][L][T^{-2}]}} \]
This simplifies to:
\[ [T] = \sqrt{\frac{[M][L]}{[M][L][T^{-2}]}} = \sqrt{[T^2]} = [T] \]

### Step 6: Final expression for dimensions of time.
Thus, we can express the dimensions of time in terms of the fundamental units as:
\[ [T] = [F^{-1/2}][M^{1/2}][L^{1/2}] \]

### Conclusion:
The dimensions of time when length, mass, and force are chosen as fundamental units are:
\[ [T] = [M^{1/2}][L^{1/2}][F^{-1/2}] \]