In a Diesel engine, the diameter of the piston is $d = 800$ mm. The pressure during the combustion phase is $20$ bar. Calculate the force exerted on the piston.

1. Problem Description

In a Diesel engine, the diameter of the piston is $d = 800$ mm. The pressure during the combustion phase is $20$ bar. Calculate the force exerted on the piston.

2. Solution Steps

First, we convert the diameter from millimeters to meters.

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

Next, we calculate the area of the piston. Since the piston is circular, the area is given by

$A = \pi r^2$, where $r$ is the radius of the piston. Since $r = d/2$, we have $r = 0.8/2 = 0.4$ m.

So,

$A = \pi (0.4)^2 = \pi(0.16) \approx 0.50265 \text{ m}^2$

Now, we convert the pressure from bar to Pascals (Pa). 1 bar is equal to $10^5$ Pa.

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

The force exerted on the piston is given by the formula

$F = P \times A$

$F = (2 \times 10^6 \text{ Pa}) \times (0.16\pi \text{ m}^2) = 2 \times 10^6 \times 0.50265 \text{ N} = 1005309.649 \text{ N} \approx 1.005 \times 10^6 \text{ N}$

3. Final Answer

The force exerted on the piston is approximately $1.005 \times 10^6$ N.