If the energy ( E) ,velocity (v) and force (F) be taken as fundamental quantities,then the dimension of mass will be

Options

A. `[Fv^(-2)]`
B. `[Fv^(-1)]`
C. `[Ev^(-2)]`
D. `[Ev^(2)]`

Explanation

To find the dimension of mass (M) when energy (E), velocity (v), and force (F) are taken as fundamental quantities, we can follow these steps:

### Step 1: Write the dimensions of energy, velocity, and force.
- **Energy (E)**: Energy is defined as work done, which is force multiplied by displacement. The dimension of work (or energy) is given by:
\[ [E] = [F] \cdot [L] \]
where \( [L] \) is the dimension of length.

- **Force (F)**: Force is defined as mass times acceleration. The dimension of force is given by:
\[ [F] = [M] \cdot [a] = [M] \cdot \left[\frac{L}{T^2}\right] \]
where \( [T] \) is the dimension of time.

- **Velocity (v)**: Velocity is defined as displacement divided by time:
\[ [v] = \frac{[L]}{[T]} \]

### Step 2: Substitute the dimensions into the equation for energy.
From the definition of energy, we can express it in terms of mass:
\[ [E] = [F] \cdot [L] = ([M] \cdot [L] \cdot [T^{-2}]) \cdot [L] = [M] \cdot [L^2] \cdot [T^{-2}] \]

### Step 3: Rearrange the equation to find the dimension of mass.
From the equation for energy, we can isolate mass:
\[ [M] = \frac{[E]}{[L^2] \cdot [T^{-2}]} \]

### Step 4: Substitute the dimension of velocity into the equation.
We know that:
\[ [v] = \frac{[L]}{[T]} \implies [L] = [v] \cdot [T] \]
Substituting this into the equation for mass:
\[ [M] = \frac{[E]}{[v^2] \cdot [T^{-2}]} \]
This can be rewritten as:
\[ [M] = [E] \cdot [T^2] \cdot [v^{-2}] \]

### Step 5: Express time (T) in terms of energy (E), force (F), and velocity (v).
From the equation for energy, we can express \( [T] \):
\[ [T] = \frac{[E]}{[F] \cdot [v]} \]
Substituting this back into the equation for mass:
\[ [M] = [E] \cdot \left(\frac{[E]}{[F] \cdot [v]}\right)^2 \cdot [v^{-2}] \]

### Step 6: Simplify the equation to find the dimension of mass.
After substituting and simplifying, we find:
\[ [M] = \frac{[E] \cdot [E]}{[F] \cdot [v^2]} = \frac{[E^2]}{[F] \cdot [v^2]} \]
This leads us to the final dimension of mass:
\[ [M] = [E][v^{-2}] \]

### Final Result:
Thus, the dimension of mass (M) in terms of energy (E), velocity (v), and force (F) is:
\[ [M] = [E][v^{-2}] \]