If force, length and time are taken as fundamental units, then the dimensions of mass will be

Options

A. `[F^(1)L^(2)T^(-2)]`
B. `[F^(1)L^(-1)T^(2)]`
C. `[F^(0)L^(1)T^(-2)]`
D. `[F^(1)L^(1)T^(-1)]`

Explanation

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

### Step-by-Step Solution:

1. **Understand the relationship between force, mass, and acceleration**:
The fundamental equation relating these quantities is given by Newton's second law:
\[ F = m \cdot a \]
where \( F \) is force, \( m \) is mass, and \( a \) is acceleration.

2. **Rearranging the equation**:
From the equation, we can express mass \( m \) in terms of force \( F \) and acceleration \( a \):
\[ m = \frac{F}{a} \]

3. **Identify the dimensions of force and acceleration**:
- The dimension of force \( F \) can be expressed in terms of mass, length, and time. Using the formula \( F = m \cdot a \), we know that:
\[ [F] = [m] \cdot [a] \]
- The dimension of acceleration \( a \) is defined as the change in velocity over time. Since velocity \( v \) is displacement over time, we have:
\[ a = \frac{v}{t} = \frac{L/T}{T} = \frac{L}{T^2} \]
Thus, the dimension of acceleration is:
\[ [a] = [L][T^{-2}] \]

4. **Substituting the dimensions**:
Now, substituting the dimensions of \( a \) back into the equation for mass:
\[ [m] = \frac{[F]}{[a]} = \frac{[F]}{[L][T^{-2}]} \]

5. **Expressing the dimension of force**:
We need to express the dimension of force \( F \) in terms of the fundamental units. Since force is defined as mass times acceleration:
\[ [F] = [m][a] = [m][L][T^{-2}] \]
Therefore, we can substitute this into our equation for mass:
\[ [m] = \frac{[m][L][T^{-2}]}{[L][T^{-2}]} \]

6. **Simplifying the expression**:
When we simplify this, we find that the dimensions of mass \( [m] \) can be expressed as:
\[ [m] = [F][L^{-1}][T^{2}] \]

7. **Final expression**:
Since we are looking for the dimensions of mass in terms of force \( F \), length \( L \), and time \( T \):
\[ [m] = [F][L^{-1}][T^{2}] \]

8. **Conclusion**:
Therefore, the dimensions of mass when force, length, and time are taken as fundamental units is:
\[ [m] = F^1 L^{-1} T^2 \]

### Answer:
The dimensions of mass are \( F^1 L^{-1} T^2 \).