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

### Step 1: Write the dimensional formula of force
The dimensional formula for force (F) is given by Newton's second law of motion:
\[ F = m \cdot a \]
where \( m \) is mass and \( a \) is acceleration. The dimensional formula for acceleration is:
\[ a = \frac{L}{T^2} \]
Thus, the dimensional formula for force can be expressed as:
\[ [F] = [M][L][T^{-2}] = M^1 L^1 T^{-2} \]

### Step 2: Express mass in terms of force, length, and time
We want to express mass (M) in terms of the fundamental units of force (F), length (L), and time (T). We can assume:
\[ M = F^a \cdot L^b \cdot T^c \]
where \( a \), \( b \), and \( c \) are the powers we need to determine.

### Step 3: Substitute the dimensional formula of force
Substituting the dimensional formula of force into the equation gives:
\[ M = (M^1 L^1 T^{-2})^a \cdot L^b \cdot T^c \]
This simplifies to:
\[ M = M^a L^{a} T^{-2a} \cdot L^b \cdot T^c \]
Combining the terms, we have:
\[ M = M^a L^{a+b} T^{-2a+c} \]

### Step 4: Equate the powers of M, L, and T
Now we equate the powers of M, L, and T from both sides:
1. For M: \( 1 = a \)
2. For L: \( 0 = a + b \)
3. For T: \( 0 = -2a + c \)

### Step 5: Solve the equations
From the first equation, we have:
\[ a = 1 \]
Substituting \( a = 1 \) into the second equation:
\[ 0 = 1 + b \implies b = -1 \]
Substituting \( a = 1 \) into the third equation:
\[ 0 = -2(1) + c \implies c = 2 \]

### Step 6: Write the final expression for mass
Now we have:
- \( a = 1 \)
- \( b = -1 \)
- \( c = 2 \)

Thus, we can write the dimensional formula for mass as:
\[ M = F^1 \cdot L^{-1} \cdot T^2 \]
This can be expressed as:
\[ M = \frac{F}{L} \cdot T^2 \]

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