A uniform plank of Young's modulus Y is moved over a smooth horizontal surface by a constant horizontal force F. The area of cross section of the plank is A. The compressive strain on the plank in the direction of the force is

Options

A. `F//AY`
B. `2F//AY`
C. `(1)/(2)(F//AY)`
D. `3F//AY`

Answer

To solve the problem, we need to determine the compressive strain on a uniform plank when a constant horizontal force \( F \) is applied to it. We will use the concept of Young's modulus and the definitions of stress and strain.

### Step-by-Step Solution:

1. **Understand Young's Modulus**:
Young's modulus \( Y \) is defined as the ratio of stress to strain in a material. Mathematically, it is given by:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]

2. **Define Stress**:
Stress (\( \sigma \)) is defined as the force (\( F \)) applied per unit area (\( A \)). Thus, we can express stress as:
\[ \sigma = \frac{F}{A} \]

3. **Define Strain**:
Strain (\( \epsilon \)) is defined as the change in length per unit original length. However, in this case, we are interested in the compressive strain due to the applied force.

4. **Relate Stress and Strain using Young's Modulus**:
From the definition of Young's modulus, we can rearrange the equation to find strain:
\[ \text{Strain} = \frac{\text{Stress}}{Y} \]

5. **Substitute Stress into the Strain Formula**:
Now, substituting the expression for stress into the strain formula:
\[ \text{Strain} = \frac{\sigma}{Y} = \frac{F/A}{Y} = \frac{F}{AY} \]

6. **Conclusion**:
Therefore, the compressive strain on the plank in the direction of the force is given by:
\[ \text{Compressive Strain} = \frac{F}{AY} \]

### Final Answer:
The compressive strain on the plank in the direction of the force is \( \frac{F}{AY} \).