If force (F), length (L) and time (T) be considered fundamenal units, then the units of mass will be

Options

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

Answer

To find the units of mass when force (F), length (L), and time (T) are considered fundamental units, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between force, mass, and acceleration**:
The formula for force 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 formula to express mass**:
We can rearrange the formula to solve for mass:
\[ m = \frac{F}{a} \]

3. **Express acceleration in terms of fundamental units**:
Acceleration (\( a \)) is defined as the change in velocity over time. Velocity (\( v \)) is distance over time:
\[ a = \frac{v}{t} = \frac{L}{T} \]
where \( L \) is length and \( T \) is time.

4. **Substituting acceleration back into the mass equation**:
Now, substituting the expression for acceleration into the mass equation:
\[ m = \frac{F}{\frac{L}{T}} = \frac{F \cdot T}{L} \]

5. **Identifying the units of force**:
The unit of force in terms of fundamental units is:
\[ F = M \cdot \frac{L}{T^2} \]
where \( M \) is mass, \( L \) is length, and \( T \) is time.

6. **Substituting the unit of force into the mass equation**:
Now, substituting the expression for force into the mass equation:
\[ m = \frac{M \cdot \frac{L}{T^2} \cdot T}{L} \]
Simplifying this gives:
\[ m = \frac{M \cdot L \cdot T}{L \cdot T^2} = \frac{M}{T} \]

7. **Final expression for mass**:
Rearranging gives:
\[ m = F \cdot T^2 \cdot L^{-1} \]
Therefore, the dimensional formula for mass in terms of the fundamental units of force, length, and time is:
\[ [M] = [F] \cdot [T^2] \cdot [L^{-1}] \]

### Conclusion:
The fundamental unit of mass can be expressed as:
\[ [M] = F \cdot T^2 \cdot L^{-1} \]