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

### Step 1: Write the formula for force
The formula for force is given by Newton's second law:
\[ F = m \cdot a \]
where \( m \) is mass and \( a \) is acceleration.

### Step 2: Express acceleration in terms of fundamental quantities
Acceleration \( a \) can be defined as the change in velocity over time:
\[ a = \frac{\Delta v}{\Delta t} \]
Velocity \( v \) is defined as displacement (length) over time:
\[ v = \frac{L}{T} \]
Thus, acceleration can be expressed as:
\[ a = \frac{L/T}{T} = \frac{L}{T^2} \]

### Step 3: Substitute acceleration back into the force equation
Substituting the expression for acceleration into the force equation gives:
\[ F = m \cdot \frac{L}{T^2} \]

### Step 4: Rearrange the equation to solve for mass
Rearranging the equation to isolate mass \( m \) gives:
\[ m = \frac{F \cdot T^2}{L} \]

### Step 5: Identify the dimensions of mass in terms of fundamental units
Now, we can express the units of mass in terms of the fundamental units of force (F), length (L), and time (T):
- The unit of force \( F \) can be expressed as \( [F] = [M][L][T^{-2}] \) (using the dimensional formula of force).
- Therefore, substituting this into the equation for mass gives:
\[ m = \frac{[M][L][T^{-2}] \cdot T^2}{L} \]

### Step 6: Simplify the expression
Simplifying this expression:
\[ m = \frac{[M][L][T^{-2}] \cdot T^2}{L} = [M] \]
This shows that the units of mass can be expressed as:
\[ [M] = [F][L^{-1}][T^2] \]

### Final Answer
Thus, the units of mass in terms of the fundamental units of force, length, and time are:
\[ [M] = [F][L^{-1}][T^2] \]