In a system of units if force (F), acceleration (A) and time (T) are taken as fundamental units, then the dimensional formula of energy will become `[FAT^(x//3)]`. Find value of x?

Answer

To find the value of \( x \) in the dimensional formula of energy expressed as \([FAT^{(x/3)}]\), we start by recalling the relationship between energy, force, acceleration, and time.

### Step 1: Understand the dimensions of energy
The dimensional formula for energy is given by:
\[ [E] = [ML^2T^{-2}] \]
where \( M \) is mass, \( L \) is length, and \( T \) is time.

### Step 2: Write the dimensions of force, acceleration, and time
1. **Force (F)** has the dimensional formula:
\[ [F] = [MLT^{-2}] \]
2. **Acceleration (A)** has the dimensional formula:
\[ [A] = [LT^{-2}] \]
3. **Time (T)** has the dimensional formula:
\[ [T] = [T] \]

### Step 3: Express energy in terms of F, A, and T
We can express energy \( E \) in terms of the fundamental units \( F \), \( A \), and \( T \):
\[ E = F^x A^y T^z \]

### Step 4: Substitute the dimensions into the equation
Substituting the dimensions of \( F \), \( A \), and \( T \) into the equation gives us:
\[ [ML^2T^{-2}] = [MLT^{-2}]^x [LT^{-2}]^y [T]^z \]

### Step 5: Expand the right-hand side
Expanding the right-hand side:
\[ [ML^2T^{-2}] = [M^x L^{x+y} T^{-2x - 2y + z}] \]

### Step 6: Equate the powers of M, L, and T
Now, we equate the powers of \( M \), \( L \), and \( T \) from both sides:
1. For \( M \):
\[ x = 1 \quad \text{(1)} \]
2. For \( L \):
\[ x + y = 2 \quad \text{(2)} \]
3. For \( T \):
\[ -2x - 2y + z = -2 \quad \text{(3)} \]

### Step 7: Solve the equations
From equation (1), we have:
\[ x = 1 \]
Substituting \( x = 1 \) into equation (2):
\[ 1 + y = 2 \implies y = 1 \]
Now substituting \( x = 1 \) and \( y = 1 \) into equation (3):
\[ -2(1) - 2(1) + z = -2 \implies -2 - 2 + z = -2 \implies z = 2 \]

### Step 8: Substitute back to find the dimensional formula of energy
Now we have:
\[ x = 1, \quad y = 1, \quad z = 2 \]
Thus, the dimensional formula of energy can be expressed as:
\[ E = F^1 A^1 T^2 \]

### Step 9: Relate it to the given expression
The problem states that the dimensional formula of energy can be expressed as:
\[ [FAT^{(x/3)}] \]
From our derived formula, we can compare:
\[ [F^1 A^1 T^2] = [F^{1} A^{1} T^{(x/3)}] \]
This implies:
\[ \frac{x}{3} = 2 \implies x = 6 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{6} \]