A uniform elastic plank moves due to a constant force `F_(0)` applied at one end whose area is `S`. The Young's modulus of the plank is `Y`. The strain produced in the direction of force is

Options

A. `F_(o)/(2SY)`
B. `F_(o)/(SY)`
C. `(2F_(0))/(SY)`
D. `sqrt(2F_(0))/(SY)`

Explanation

To find the strain produced in the direction of the force applied to a uniform elastic plank, we will follow these steps:

### Step 1: Understand the relationship between stress, strain, and Young's modulus
The relationship between stress (σ), strain (ε), and Young's modulus (Y) is given by the formula:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]
Where:
- Stress (σ) = \(\frac{F_0}{S}\) (Force per unit area)
- Strain (ε) = \(\frac{\Delta L}{L}\) (Change in length per original length)

### Step 2: Calculate the stress applied to the plank
The stress applied to the plank due to the force \(F_0\) is:
\[ \sigma = \frac{F_0}{S} \]

### Step 3: Relate stress to strain using Young's modulus
From the definition of Young's modulus, we can express strain in terms of stress:
\[ \epsilon = \frac{\sigma}{Y} \]
Substituting the expression for stress:
\[ \epsilon = \frac{F_0/S}{Y} \]

### Step 4: Simplify the expression for strain
Now, we can simplify the expression:
\[ \epsilon = \frac{F_0}{SY} \]

### Step 5: Final expression for strain
Thus, the strain produced in the direction of the force is:
\[ \epsilon = \frac{F_0}{SY} \]

### Conclusion
The strain produced in the direction of the force applied to the uniform elastic plank is given by:
\[ \epsilon = \frac{F_0}{SY} \]