If force F, area A and density D are teken as the fundamental units, the representation of Young's modulus 'Y' will be

Options

A. `[F^(-1)A^(-1)D^(-1)]`
B. `[FA^(-2)D^(2)]`
C. `[FA^(-1)D]`
D. `[FA^(-1)D^(0)]`

Answer

To find the representation of Young's modulus \( Y \) when force \( F \), area \( A \), and density \( D \) are taken as fundamental units, we can follow these steps:

### Step 1: Understand Young's Modulus
Young's modulus \( Y \) is defined as the ratio of stress to strain. Mathematically, it can be expressed as:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]

### Step 2: Define Stress and Strain
- **Stress** is defined as force per unit area:
\[ \text{Stress} = \frac{F}{A} \]
- **Strain** is a dimensionless quantity, defined as the change in length divided by the original length:
\[ \text{Strain} = \frac{\Delta L}{L} \]
Since strain is dimensionless, it does not contribute to the dimensions of Young's modulus.

### Step 3: Substitute Stress into Young's Modulus
Substituting the expression for stress into the formula for Young's modulus gives:
\[ Y = \frac{F/A}{\text{Strain}} = \frac{F}{A} \]

### Step 4: Determine the Dimensions of Force and Area
- The dimension of force \( F \) is given by:
\[ [F] = MLT^{-2} \]
where \( M \) is mass, \( L \) is length, and \( T \) is time.
- The dimension of area \( A \) is:
\[ [A] = L^2 \]

### Step 5: Calculate the Dimensions of Young's Modulus
Now substituting the dimensions of force and area into the expression for Young's modulus:
\[ Y = \frac{[F]}{[A]} = \frac{MLT^{-2}}{L^2} = M L^{-1} T^{-2} \]

### Step 6: Express in Terms of Fundamental Units
Since we are considering force \( F \), area \( A \), and density \( D \) as fundamental units, we need to express Young's modulus in terms of these units:
- The dimension of density \( D \) is:
\[ [D] = \frac{M}{L^3} \]
This means that density does not contribute to the dimensions of Young's modulus since it is dimensionless in this context.

### Final Representation
Thus, the representation of Young's modulus \( Y \) in terms of the fundamental units of force \( F \), area \( A \), and density \( D \) is:
\[ Y = F^1 A^{-1} D^0 \]
This simplifies to:
\[ Y = F^1 A^{-1} \]

### Conclusion
Therefore, the representation of Young's modulus \( Y \) when force, area, and density are taken as fundamental units is:
\[ Y = F^1 A^{-1} D^0 \]