The shear modulus of steel is `8.1xx10^(10)N//m^(2)`. A steel nail of radius `7.5xx10^(-4)m` projects 0.040 m horizontally outward from a wall. A man hangs a wet raincoat of weight 28.5 N from the end of the nail.Assuming the wall holds its end of the nail, what is the vertical deflection of the other end of the nail ?

Options

A. `1.8xx10^(-3)m`
B. `3.3xx10^(-2)m`
C. `7.9xx10^(-6)m`
D. `4.2xx10^(-4)m`

Answer

To solve the problem of finding the vertical deflection of the steel nail when a weight is hung from it, we can follow these steps:

### Step 1: Identify the Given Values
- Shear modulus of steel, \( G = 8.1 \times 10^{10} \, \text{N/m}^2 \)
- Radius of the nail, \( r = 7.5 \times 10^{-4} \, \text{m} \)
- Length of the nail, \( L = 0.04 \, \text{m} \)
- Weight of the raincoat, \( F = 28.5 \, \text{N} \)

### Step 2: Calculate the Cross-Sectional Area of the Nail
The cross-sectional area \( A \) of the nail can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
Substituting the value of the radius:
\[ A = \pi (7.5 \times 10^{-4})^2 \approx 3.14 \times (5.625 \times 10^{-7}) \approx 1.767 \times 10^{-6} \, \text{m}^2 \]

### Step 3: Use the Shear Modulus Formula
The shear modulus \( G \) is defined as:
\[ G = \frac{F/A}{\Delta y / L} \]
Rearranging this formula to solve for the vertical deflection \( \Delta y \):
\[ \Delta y = \frac{F \cdot L}{A \cdot G} \]

### Step 4: Substitute the Values into the Formula
Now, substituting the known values into the equation:
\[ \Delta y = \frac{28.5 \, \text{N} \cdot 0.04 \, \text{m}}{1.767 \times 10^{-6} \, \text{m}^2 \cdot 8.1 \times 10^{10} \, \text{N/m}^2} \]

### Step 5: Calculate the Deflection
Calculating the numerator:
\[ 28.5 \times 0.04 = 1.14 \, \text{N m} \]
Calculating the denominator:
\[ 1.767 \times 10^{-6} \times 8.1 \times 10^{10} \approx 1.432 \times 10^5 \, \text{N} \]
Now, substituting these values back:
\[ \Delta y = \frac{1.14}{1.432 \times 10^5} \approx 7.96 \times 10^{-6} \, \text{m} \]

### Step 6: Final Result
Thus, the vertical deflection of the other end of the nail is approximately:
\[ \Delta y \approx 7.96 \times 10^{-6} \, \text{m} \]