The problem asks us to calculate the force exerted on the piston of a diesel engine during combustion. The diameter of the piston is given as $d = 800 \text{ mm}$, and the pressure during combustion is $P = 20 \text{ bar}$. We are provided with the formula $A = \frac{\pi \cdot d^2}{4}$ which represents the area of the piston.

1. Problem Description

The problem asks us to calculate the force exerted on the piston of a diesel engine during combustion. The diameter of the piston is given as $d = 800 \text{ mm}$, and the pressure during combustion is $P = 20 \text{ bar}$. We are provided with the formula $A = \frac{\pi \cdot d^2}{4}$ which represents the area of the piston.

2. Solution Steps

First, we need to convert the diameter from millimeters to meters:

$d = 800 \text{ mm} = 0.8 \text{ m}$.

Next, calculate the area of the piston using the given formula:

$A = \frac{\pi \cdot d^2}{4} = \frac{\pi \cdot (0.8)^2}{4} = \frac{\pi \cdot 0.64}{4} = 0.16\pi \text{ m}^2$.

Now we need to convert the pressure from bar to Pascals (Pa), where 1 bar = $10^5$ Pa:

$P = 20 \text{ bar} = 20 \cdot 10^5 \text{ Pa} = 2 \cdot 10^6 \text{ Pa}$.

The force $F$ exerted on the piston is given by the formula:

$F = P \cdot A$.

Substituting the values we have:

$F = (2 \cdot 10^6 \text{ Pa}) \cdot (0.16\pi \text{ m}^2) = 0.32\pi \cdot 10^6 \text{ N} = 320000\pi \text{ N}$.

Approximating $\pi$ as $3.14159$:

$F \approx 320000 \cdot 3.14159 \text{ N} \approx 1005308.8 \text{ N}$.

3. Final Answer

The force exerted on the piston is approximately $1005308.8$ N.