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

Options

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

Answer

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**:
We start with the relationship that mass (m) can be expressed in terms of force (F), length (L), and time (T). We can write:
\[ m \propto F^a L^b T^c \]
where \( a, b, c \) are the powers to which force, length, and time are raised, respectively.

2. **Use the Dimensional Formula for Force**:
The dimensional formula for force is given by:
\[ F = MLT^{-2} \]
Here, M is mass, L is length, and T is time.

3. **Express Mass in Terms of Dimensions**:
Substituting the dimensions of force into our equation, we have:
\[ m = k (MLT^{-2})^a L^b T^c \]
This simplifies to:
\[ m = k M^a L^{a+b} T^{-2a+c} \]

4. **Set Up the Dimensional Equation**:
Since mass has the dimension \( M^1 L^0 T^0 \), we can equate the dimensions:
\[ M^1 L^0 T^0 = M^a L^{a+b} T^{-2a+c} \]

5. **Compare the Exponents**:
From the above equation, we can compare the exponents of M, L, and T:
- For M: \( a = 1 \)
- For L: \( a + b = 0 \)
- For T: \( -2a + c = 0 \)

6. **Solve the Equations**:
- From \( a = 1 \), we substitute into the second equation:
\[ 1 + b = 0 \implies b = -1 \]
- Now substitute \( a = 1 \) into the third equation:
\[ -2(1) + c = 0 \implies c = 2 \]

7. **Final Dimensions of Mass**:
Now we have:
- \( a = 1 \)
- \( b = -1 \)
- \( c = 2 \)

Therefore, the dimensions of mass can be expressed as:
\[ [m] = F^1 L^{-1} T^2 \]

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