Treating force F, length L and time T as fundamental units, what is the dimensional formula of mass?

Options

A. `F^(+1)L^(+2)T^(+1)`
B. `F^(+2)L^(+1)T^(-1)`
C. `F^(+1)L^(-1)T^(+2)`
D. `F^(+1)L^(+1)T^(+2)`

Explanation

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

### Step 1: Understand the relationship between force, mass, and acceleration
The equation for force is given by Newton's second law:
\[ F = m \cdot a \]
where:
- \( F \) is the force,
- \( m \) is the mass,
- \( a \) is the acceleration.

### Step 2: Express acceleration in terms of fundamental units
Acceleration \( a \) is defined as the rate of change of velocity. Velocity \( v \) is defined as displacement (length) over time:
\[ a = \frac{v}{t} \]
Since \( v = \frac{L}{T} \), we can substitute this into the equation for acceleration:
\[ a = \frac{L}{T^2} \]

### Step 3: Substitute acceleration back into the force equation
Now we can rewrite the force equation using the expression for acceleration:
\[ F = m \cdot \frac{L}{T^2} \]
Rearranging this gives us:
\[ m = \frac{F \cdot T^2}{L} \]

### Step 4: Determine the dimensional formula for mass
Now we need to express the dimensional formula for mass in terms of the dimensions of force, length, and time. The dimensional formula for force \( F \) can be expressed as:
\[ [F] = [M][L][T^{-2}] \]
where \( [M] \) is the dimensional formula for mass.

Substituting this into our equation for mass:
\[ m = \frac{[M][L][T^{-2}] \cdot T^2}{L} \]
This simplifies to:
\[ m = [M] \]

### Step 5: Rearranging the equation
From the force equation, we can express the dimensional formula of mass as:
\[ [M] = \frac{[F][T^2]}{[L]} \]
Substituting the dimensional formula for force:
\[ [M] = \frac{[M][L][T^{-2}] \cdot T^2}{L} \]
This simplifies to:
\[ [M] = [F][L^{-1}][T^{2}] \]

### Step 6: Final dimensional formula
Thus, we can express the dimensional formula of mass in terms of force, length, and time:
\[ [M] = [F^{1}][L^{-1}][T^{2}] \]

### Final Answer
The dimensional formula of mass (M) in terms of force (F), length (L), and time (T) is:
\[ [M] = F^{1} L^{-1} T^{2} \]

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