Boyle's law pressure calculation

[Daan van Driel]

Homework Statement

An pillow is inflated with an pressure of let's say 20 hPa. Afterwards we place a weight of 10 Kg on the pillow, what is the pressure increase within the pillow?

P athmosthere = 1013 hPa
P cushion = 1013 + 20 = 1033 hPa
m = 10 Kg
g = 9.81 m/s^2

Amount of air in the pillow stays the same

Homework Equations

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Boyle's law: P1 * V1 = P2 * V2
Gravitational force: F = m * g
Pressure: P = F / A

The Attempt at a Solution

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The added weight creates a force F. This creates an apposing force within the pillow (Newton's third law) which is devided over the enitre inner surface of the pillow. Therefore P = F / A = m * g / A.

Lets assume that the surface area of the pillow is equal to 300 cm^2. P = (10 * 9.81) / 0.03 = 3270

Would this be right? If so, how do I translate this 3270 back to the pressure difference between P2 ane P3?

 

[Chestermiller]

Is this an actual homework problem? Is the geometry exactly as you have shown it in the figure?

 

[Daan van Driel]

Is this an actual homework problem? Is the geometry exactly as you have shown it in the figure?

No, this is a problem I'm trying to answer for a project. The geometry is not exactly as it is shown in the picture, i just made it to substantiate my explanation.

 

[Chestermiller]

No, this is a problem I'm trying to answer for a project. The geometry is not exactly as it is shown in the picture, i just made it to substantiate my explanation.

This is a pretty complicated problem, since the shape and curvature of your surface changes. You also omitted the force on the other side of the cushion required to hold it in place. It might be possible to model this more easily if the cushion were sandwiched between two infinite flat surfaces, but that too would be somewhat complicated. If the outer boundary of the cushion were considered inextensible, then pressing it between flat surfaces (or any other surfaces for that matter) would cause the boundary to buckle and form folds.