To solve this problem, we need to calculate the force applied on the piston given its diameter and the pressure in the cylinder. Let's go through the steps:
1. Given Values:
- Diameter of the piston: [tex]\( d = 1.4 \)[/tex] meters.
- Pressure in the cylinder: [tex]\( P = 4.0 \times 10^5 \)[/tex] Pascals (Pa).
2. Calculate the Radius:
The radius [tex]\( r \)[/tex] is half of the diameter. So,
[tex]\[
r = \frac{d}{2} = \frac{1.4 \, \text{m}}{2} = 0.7 \, \text{m}
\][/tex]
3. Calculate the Area of the Piston:
The piston is circular, so the area [tex]\( A \)[/tex] can be calculated using the formula for the area of a circle:
[tex]\[
A = \pi r^2
\][/tex]
Substituting the radius:
[tex]\[
A = \pi (0.7 \, \text{m})^2
\][/tex]
Numerical calculation yields:
[tex]\[
A \approx 1.539 \, \text{m}^2
\][/tex]
4. Calculate the Force:
Using the formula [tex]\( \text{Force} = \text{Pressure} \times \text{Area} \)[/tex]:
[tex]\[
F = P \times A
\][/tex]
Substituting the pressure and the area:
[tex]\[
F = 4.0 \times 10^5 \, \text{Pa} \times 1.539 \, \text{m}^2
\][/tex]
Numerical calculation yields:
[tex]\[
F \approx 615752 \, \text{N}
\][/tex]
Thus, the force applied on the piston is approximately 615752.16 Newtons (N).
