Physics of spring airgun cylinder system?

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The discussion centers on understanding the physics behind spring airgun cylinder systems, particularly the calculations for pressure when a force of 550 Newtons is applied. The user seeks to grasp the necessary math despite lacking a mechanical engineering background. Key points include the system's total volume of 37,464 mm³, the piston volume of 24,543 mm³, and the barrel volume of 12,921 mm³. It is noted that if the 550N force is at the end of the stroke, pressure can be calculated by dividing the force by the piston area. The user is open to further clarification or details needed for accurate calculations.
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So recently I became more interested in CAD and i found myself drafting designs for spring airguns cylinders ect. All of-course meaningless given that I have zero understanding of the math behind the design. I am no Mechanical engineer, I am a business student so if I am getting into advanced stuff here let me know but I still intend on giving it my best..

/end rant/

Essentially I would like to understand how to calculate the pressure in a system when a Force of 550 Newtons is applied to it.

System volume: 37464
Piston(Right Cylinder) Volume: 24543 mm
Barrel(Right Cylinder) Volume: 12921mm
Change in Volume at piston peak: 24543

The gas inside the system is at sea level atmospheric pressure

I have also attached an image for reference!

if I am leaving anything out please let me know and Ill attempt to elaborate more!
ip47l2.jpg
 
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If the 550N force is the force at the end of the stroke then the pressure is simply force divided by piston area.
 

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