Consider a 2 cm thick steel plate with an ultimate shear stress of `3.4 xx 10^8 Nm^(-2)`. Now, to punch a hole of diameter 2.8 cm on this plate, a minimum force of F kN is used. What is `F/10 = ?`

Explanation

To solve the problem of determining the minimum force \( F \) required to punch a hole in a steel plate, we can follow these steps:

### Step 1: Understand the Problem
We have a steel plate that is 2 cm thick with an ultimate shear stress of \( 3.4 \times 10^8 \, \text{N/m}^2 \). We need to punch a hole with a diameter of 2.8 cm. Our goal is to find the value of \( \frac{F}{10} \).

### Step 2: Calculate the Area of the Hole
The area \( A \) of the circular hole can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
where \( r \) is the radius of the hole. The diameter of the hole is 2.8 cm, so the radius \( r \) is:
\[ r = \frac{2.8 \, \text{cm}}{2} = 1.4 \, \text{cm} = 1.4 \times 10^{-2} \, \text{m} \]
Now, substituting the radius into the area formula:
\[ A = \pi (1.4 \times 10^{-2})^2 = \pi (1.96 \times 10^{-4}) \approx 6.15752 \times 10^{-4} \, \text{m}^2 \]

### Step 3: Relate Force to Shear Stress
The force \( F \) required to punch the hole can be related to the ultimate shear stress \( \tau \) and the area \( A \) using the equation:
\[ F = \tau \cdot A \]
Given that the ultimate shear stress \( \tau = 3.4 \times 10^8 \, \text{N/m}^2 \), we can substitute the values:
\[ F = (3.4 \times 10^8) \cdot (6.15752 \times 10^{-4}) \]

### Step 4: Calculate the Force
Now, performing the multiplication:
\[ F \approx 3.4 \times 10^8 \times 6.15752 \times 10^{-4} \approx 209,339.68 \, \text{N} \]
Converting this to kilonewtons:
\[ F \approx 209.34 \, \text{kN} \]

### Step 5: Find \( \frac{F}{10} \)
Now, we need to calculate \( \frac{F}{10} \):
\[ \frac{F}{10} \approx \frac{209.34}{10} \approx 20.934 \]
Rounding this gives:
\[ \frac{F}{10} \approx 20.93 \]

### Final Answer
Thus, the final answer is:
\[ \frac{F}{10} \approx 20.93 \]