If force `(F)`, velocity `(V)` and time `(T)` are taken as fundamental units, then the dimensions of mass are

Options

A. `["FvT"^(-1)]`
B. `["FvT"^(-2)]`
C. `["Fv"^(-1)"T"^(-1)]`
D. `["Fv"^(-1)"T"]`

Explanation

To find the dimensions of mass when force (F), velocity (V), and time (T) are taken as fundamental units, we can start by recalling Newton's second law of motion, which states that:

\[ F = m \cdot a \]

where \( m \) is mass and \( a \) is acceleration.

### Step 1: Express acceleration in terms of fundamental units
Acceleration \( a \) can be defined as the change in velocity over time, which can be mathematically expressed as:

\[ a = \frac{V}{T} \]

### Step 2: Rearrange the equation for mass
From the equation of force, we can express mass \( m \) in terms of force and acceleration:

\[ m = \frac{F}{a} \]

Substituting the expression for acceleration from Step 1:

\[ m = \frac{F}{\frac{V}{T}} \]

### Step 3: Simplify the expression for mass
This can be rewritten as:

\[ m = \frac{F \cdot T}{V} \]

### Step 4: Write the dimensions of mass
Now, we can express the dimensions of mass in terms of the dimensions of force, velocity, and time:

The dimensional formula for mass can be expressed as:

\[ [m] = [F] \cdot [T] \cdot [V]^{-1} \]

### Step 5: Substitute the dimensions of force and velocity
Using the known dimensions:
- The dimension of force \( [F] \) is \( [M][L][T]^{-2} \)
- The dimension of velocity \( [V] \) is \( [L][T]^{-1} \)

Thus, substituting these into our expression for mass:

\[ [m] = [F] \cdot [T] \cdot [V]^{-1} = \left([M][L][T]^{-2}\right) \cdot [T] \cdot \left([L][T]^{-1}\right)^{-1} \]

### Step 6: Simplify the dimensional expression
This simplifies to:

\[ [m] = [M][L][T]^{-2} \cdot [T] \cdot [L]^{-1}[T]^{1} = [M][L]^{0}[T]^{-1} \]

### Final Expression
Thus, the dimensions of mass in terms of the fundamental units of force, velocity, and time is:

\[ [m] = F \cdot V^{-1} \cdot T \]

### Conclusion
The correct answer is:

**Option: FV^{-1}T**

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