How Do I Calculate Pressure Changes in a Linear Spring System?

[Saladsamurai]

Homework Statement

It's 2.31

Here is what I am thinking so far:

At V2 the system is at equilibrium, so p2 equals p_atm since the spring exerts no force.

So p2 is known.

I now need p1. I am not sure where to go from here. I have V1, V2 and p2.
I think that spring-force/area is part of p1...

Can someone give me a hint? Just a hint...

 

[edziura]

The work done by the piston on the air is

$$\int$$ P dV

You could write a linear equation for P as a fcn of V with the info you have, but of course geometrically this is just the area under the graph of P vs V, which in this case is trapezoidal in shape. Sketch the P-V trapezoid, calculate its area (watch the units), and you gave the work.

 

[Saladsamurai]

That sounds reasonable! But, let's look at it for a moment w/out the p-V diagram, I am confused as to how to right out $\int pdV$ What kind of compression is this considered? p is not constant, nor is V. I am just confused as to how to incorporate the spring force?

 

[edziura]

It says the spring force varies (decreases) linearly as the volume. You have two sets of (P,V) graph points, so you could write an equation of the form P = mV + c and use that to calculate the integral.

 

[Saladsamurai]

I wish I followed you. What is P=mV+c ? It say that the spring force varies linearly, not the pressure...

 

[Saladsamurai]

Are you saying that p can be written as a function of volume since ot is just the sum of p_atm and F_spring/area?

That is that $$p(V)=F/A+p_{atm}$$ so I have to find out what F is...

So I need to write F_spring as a function of V somehow...

 

[edziura]

We know the following:

The spring force varies linearly from 9.00 kN to 0 kN. I'm writng this in kN for convenience, since kN / m^2 is kPa. Since the area of the piston is constant, and P = F / A, the pressure varies linearly also (as volume decreases) and so we can write a function P(V) = mV + c.

Now, when V = 0.003 m^3, F = 9.0 kN and P = (9.0 kN / 0.018 m^2) + 100 kPa = 600 kPa

When V = 0.002 m^3, F = 0.0 kN and P = (0.0 kN / 0.018 m^2) + 100 kPa = 100 kPa

Now using the two points (P,V) = (600,0.003), (100,0.002), we can find m and c for our function P(V).