In a system of units if force (F), acceleration (A) and time (T) are taken as fundamental units, then the dimensional formula of energy is

Options

A. `FA^2T`
B. `FAT^2`
C. `F^2AT`
D. `FAT`

Explanation

To find the dimensional formula of energy when force (F), acceleration (A), and time (T) are taken as fundamental units, we can follow these steps:

### Step 1: Understand the relationship of energy
Energy can be expressed in terms of force and displacement. The formula for energy is given by:
\[ \text{Energy} = \text{Force} \times \text{Displacement} \]

### Step 2: Write the dimensional formula for force and displacement
1. The dimensional formula for force (F) is:
\[ [F] = [M][L][T^{-2}] \]
where \(M\) is mass, \(L\) is length, and \(T\) is time.

2. The dimensional formula for displacement (which is length) is:
\[ [L] = [L] \]

### Step 3: Combine the dimensional formulas
Now, substituting the dimensional formulas into the energy equation:
\[ [E] = [F] \times [L] = [M][L][T^{-2}] \times [L] = [M][L^2][T^{-2}] \]

### Step 4: Express energy in terms of fundamental units
Since we are considering force (F), acceleration (A), and time (T) as fundamental units, we need to express energy in terms of these units. We can assume:
\[ [E] \propto [F^X][A^Y][T^Z] \]

### Step 5: Write the dimensional formulas for acceleration and time
1. The dimensional formula for acceleration (A) is:
\[ [A] = [L][T^{-2}] \]

2. The dimensional formula for time (T) is simply:
\[ [T] = [T] \]

### Step 6: Set up the equation
Now substituting the dimensional formulas into the energy equation:
\[ [E] = [F^X][A^Y][T^Z] = [M^X][L^{X + Y}][T^{-2X - 2Y + Z}] \]

### Step 7: Compare the powers of M, L, and T
Now we can compare the powers of M, L, and T from both sides:
1. For mass (M):
\[ X = 1 \]
2. For length (L):
\[ X + Y = 2 \]
3. For time (T):
\[ -2X - 2Y + Z = -2 \]

### Step 8: Solve the equations
From the first equation, we have \(X = 1\). Substituting \(X\) into the second equation:
\[ 1 + Y = 2 \implies Y = 1 \]
Now substituting \(X\) and \(Y\) into the third equation:
\[ -2(1) - 2(1) + Z = -2 \implies -2 - 2 + Z = -2 \implies Z = 2 \]

### Step 9: Write the final dimensional formula for energy
Now we have:
- \(X = 1\)
- \(Y = 1\)
- \(Z = 2\)

Thus, the dimensional formula for energy in terms of F, A, and T is:
\[ [E] = F^1 A^1 T^2 \]

### Final Answer
The dimensional formula of energy is:
\[ E = F \cdot A \cdot T^2 \]

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