If` "force" F, "length" L` and `time T `are taken as fundemental unit , the dimensional formula mass will be

Options

A. `[FL^(-1)T^(2)]`
B. `[FLT^(-2)]`
C. `[FL^(-1)T^(-1)]`
D. `[FL^(-5)T^(2)]`

Explanation

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

### Step 1: Understand the relationship between force, mass, and acceleration.
We know from Newton's second law of motion that:
\[ F = m \cdot a \]
where:
- \( F \) is the force,
- \( m \) is the mass,
- \( a \) is the acceleration.

### Step 2: Express acceleration in terms of fundamental units.
Acceleration (a) can be expressed as:
\[ a = \frac{v}{t} \]
where \( v \) is velocity and \( t \) is time. Velocity can be expressed as:
\[ v = \frac{L}{T} \]
Thus, we can rewrite acceleration as:
\[ a = \frac{L/T}{T} = \frac{L}{T^2} \]

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

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

### Step 5: Write the dimensional formula for mass.
Now we need to express the dimensional formula for mass in terms of the fundamental units F, L, and T.

From the equation:
\[ m = \frac{F \cdot T^2}{L} \]

We can express the dimensional formula of mass (M) as:
\[ [M] = [F] \cdot [T^2] \cdot [L^{-1}] \]

### Step 6: Assign dimensions to force.
The dimensional formula for force (F) is:
\[ [F] = [M][L][T^{-2}] \]
where [M] is the dimension of mass, [L] is the dimension of length, and [T] is the dimension of time.

### Step 7: Substitute the dimensional formula of force into the mass formula.
Substituting the dimensional formula of force into the mass formula gives us:
\[ [M] = [M][L][T^{-2}] \cdot [T^2] \cdot [L^{-1}] \]

### Step 8: Simplify the expression.
Now simplifying:
\[ [M] = [M][L][T^{-2}] \cdot [T^2] \cdot [L^{-1}] = [M][L^0][T^0] = [M] \]

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