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^(2)T`
B. `FAT^(2)`
C. `FA^(2)T^(3)`
D. `FAT`

Answer

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-by-Step Solution:

1. **Understand the relationship of energy**:
Energy (E) can be expressed in terms of force (F), acceleration (A), and time (T). The general form can be written as:
\[ E = k \cdot F^a \cdot A^b \cdot T^c \]
where \( k \) is a constant and \( a, b, c \) are the powers we need to determine.

2. **Write the dimensional formulas**:
- The dimensional formula for force (F) is:
\[ [F] = MLT^{-2} \]
- The dimensional formula for acceleration (A) is:
\[ [A] = LT^{-2} \]
- The dimensional formula for time (T) is:
\[ [T] = T \]

3. **Substitute the dimensional formulas into the energy equation**:
Substitute the dimensional formulas into the equation for energy:
\[ [E] = [F]^a \cdot [A]^b \cdot [T]^c = (MLT^{-2})^a \cdot (LT^{-2})^b \cdot (T)^c \]
This expands to:
\[ [E] = M^a \cdot L^{a+b} \cdot T^{-2a - 2b + c} \]

4. **Set up the equations based on dimensional analysis**:
Since energy has the dimensional formula of \( ML^2T^{-2} \), we can equate the powers of M, L, and T:
- For mass (M):
\[ a = 1 \]
- For length (L):
\[ a + b = 2 \]
- For time (T):
\[ -2a - 2b + c = -2 \]

5. **Solve the equations**:
From \( a = 1 \):
- Substitute \( a \) into \( a + b = 2 \):
\[ 1 + b = 2 \implies b = 1 \]
- Substitute \( a \) and \( b \) into \( -2a - 2b + c = -2 \):
\[ -2(1) - 2(1) + c = -2 \implies -2 - 2 + c = -2 \implies c = 2 \]

6. **Write the final expression for energy**:
Now we have \( a = 1 \), \( b = 1 \), and \( c = 2 \). Therefore, the dimensional formula for energy can be expressed as:
\[ E = F^1 \cdot A^1 \cdot T^2 = FAT^2 \]

### Conclusion:
The dimensional formula of energy in terms of the fundamental units of force, acceleration, and time is:
\[ E = FAT^2 \]