If velocity v acceleration A and force F are chosen as fundamental quantities, then the dimensional formula of angular momentum is terms of v,A and F would be

Options

A. `FA^(1)v`
B. `Fv^(3)A^(-2)`
C. `Fv^(2)A^(-1)`
D. `F^(2)v^(2)A^(-1)`

Explanation

To find the dimensional formula of angular momentum in terms of velocity (v), acceleration (A), and force (F), we start with the known dimensional formula of angular momentum and then express it in terms of the given quantities.

### Step-by-step Solution:

1. **Identify the Dimensional Formula of Angular Momentum**:
Angular momentum (L) is defined as the product of the moment of inertia and angular velocity. The dimensional formula for angular momentum is:
\[ [L] = [M][L^2][T^{-1}] \]
where \(M\) is mass, \(L\) is length, and \(T\) is time.

2. **Express the Given Quantities in Terms of Dimensions**:
- Velocity (v) has the dimensional formula:
\[ [v] = [L][T^{-1}] \]
- Acceleration (A) has the dimensional formula:
\[ [A] = [L][T^{-2}] \]
- Force (F) has the dimensional formula:
\[ [F] = [M][L][T^{-2}] \]

3. **Set Up the Dimensional Equation**:
We can express angular momentum in terms of the chosen fundamental quantities:
\[ [L] = [v^x][A^y][F^z] \]
Substituting the dimensional formulas we have:
\[ [M^0][L^2][T^{-1}] = ([L][T^{-1}])^x \cdot ([L][T^{-2}])^y \cdot ([M][L][T^{-2}])^z \]

4. **Expand and Combine the Dimensions**:
Expanding the right-hand side gives:
\[ [M^0][L^2][T^{-1}] = [L^x][T^{-x}] \cdot [L^y][T^{-2y}] \cdot [M^z][L^z][T^{-2z}] \]
Combining the dimensions, we have:
\[ [M^z][L^{x+y+z}][T^{-x-2y-2z}] \]

5. **Equate the Exponents**:
Now, we can equate the exponents of \(M\), \(L\), and \(T\):
- For mass (M):
\[ z = 0 \]
- For length (L):
\[ x + y + z = 2 \]
- For time (T):
\[ -x - 2y - 2z = -1 \]

6. **Solve the Equations**:
From \(z = 0\), we substitute into the other equations:
- From \(x + y = 2\)
- From \(-x - 2y = -1\)

Rearranging the second equation gives:
\[ x + 2y = 1 \]

Now we can solve these two equations:
- From \(x + y = 2\) (1)
- From \(x + 2y = 1\) (2)

Subtract (1) from (2):
\[ (x + 2y) - (x + y) = 1 - 2 \implies y = -1 \]
Substituting \(y = -1\) into (1):
\[ x - 1 = 2 \implies x = 3 \]
Thus, we find:
\[ x = 3, \quad y = -1, \quad z = 0 \]

7. **Write the Final Expression**:
Substituting \(x\), \(y\), and \(z\) back into the expression for angular momentum:
\[ [L] = [F^0][v^3][A^{-1}] = v^3 A^{-1} \]

### Final Answer:
The dimensional formula of angular momentum in terms of velocity (v), acceleration (A), and force (F) is:
\[ L = v^3 A^{-1} \]