The problem asks to calculate the force exerted on a piston in a diesel engine. The diameter of the piston is given as $d = 800 \, \text{mm}$, and the pressure during combustion is $P = 20 \, \text{bar}$. The area formula given is $A = \frac{\pi d^2}{y}$ However, judging by the context and the problem's objective (calculating the force on a circular piston given pressure), $y$ must be $4$. Therefore, the correct area formula is $A = \frac{\pi d^2}{4}$.

1. Problem Description

The problem asks to calculate the force exerted on a piston in a diesel engine. The diameter of the piston is given as $d = 800 \, \text{mm}$, and the pressure during combustion is $P = 20 \, \text{bar}$. The area formula given is $A = \frac{\pi d^2}{y}$

However, judging by the context and the problem's objective (calculating the force on a circular piston given pressure), $y$ must be $4$. Therefore, the correct area formula is $A = \frac{\pi d^2}{4}$.

2. Solution Steps

First, convert the diameter from millimeters to meters:

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

Next, convert the pressure from bar to Pascals (Pa):

$1 \, \text{bar} = 10^5 \, \text{Pa}$

$P = 20 \, \text{bar} = 20 \times 10^5 \, \text{Pa} = 2 \times 10^6 \, \text{Pa}$

Calculate the area of the piston using the formula $A = \frac{\pi d^2}{4}$:

$A = \frac{\pi (0.8 \, \text{m})^2}{4} = \frac{\pi (0.64 \, \text{m}^2)}{4} = 0.16 \pi \, \text{m}^2 \approx 0.5027 \, \text{m}^2$

The force $F$ is calculated using the formula $F = P \times A$:

$F = (2 \times 10^6 \, \text{Pa}) \times (0.16 \pi \, \text{m}^2) = 3.2 \pi \times 10^5 \, \text{N} \approx 1005309.6 \, \text{N}$

$F \approx 1.005 \times 10^6 \, \text{N}$

3. Final Answer

The force exerted on the piston is approximately $1005309.6 \, \text{N}$, which is about $1.005 \times 10^6 \, \text{N}$.