Insulated cylinder and compressing piston....

[fabianprewett]

Homework Statement

Given a thermally insulated right circular cylinder. A piston is on top with weight Wp. Sand is placed on top of the piston with weight Ws. The pressure is only dependent on the weight of the piston and sand. The molar heat capacities are constant.

a) Find the final volume when the sand is placed grain by grain.

b) Find the final volume when the sand is placed all at once.

c) Find the work done for part a) and b)

2. Homework Equations
gamma=y
A=area of piston
P1=Wp/A
P2=(Wp+Ws)/A
PV=nRT
P1V1^y=P2V2^y

The Attempt at a Solution

I got the solution for part a) using PV^gamma. I get V1/V2=(P2/P1)^(1/y)

Then plug in P2,P1,V1 I get V2=(nRtA/Wp)(Wp/(Wp+Ws))^(1/y)

For part b) I feel like I'm missing some fundamental knowledge or step to find the volume. I don't understand why the volume would be different even though I know it is. I got the equations for work and don't have a problem with part c) but obviously I need the volume in part b) to find the work.

 

[haruspex]

I don't understand why the volume would be different

Consider the height at which each grain is placed.

 

[fabianprewett]

I know P1V1^y=P2V2^y can't be used for part b) as in part a). This is what I'm missing. I don't know how to find the relationship with gamma and volume for part b).

 

[haruspex]

I don't know how to find the relationship with gamma and volume for part b).

Don't worry about that for the moment. Think about what is being done differently with the sand. The sand grains descend, but do they all descend by the same distance?

 

[fabianprewett]

I know the process is now irreversible in b). In a) it is a quasistatic, reversible process. I'm unsure of what I'm trying to find if not a relationship like P1V1^y=P2V2^y as in part a). Can part b) not be solved similarly to part a)?

 

[haruspex]

Can part b) not be solved similarly to part a)?

No, because the formula used in a) assumes the work done is only that needed to raise the pressure to the new level.
Please, please try to answer my question in post #4.

 

[fabianprewett]

The sand moves down all at once in part b). So yes they move the same distance.

 

[haruspex]

The sand moves down all at once in part b). So yes they move the same distance.

In part b) all the sand grains move the same distance, but do they in part a)?

 

[fabianprewett]

No but I've already solved everything for a). Do I need to use information from part a) to solve b)? I should be able to solve for b) indepedant of a)

 

[haruspex]

I've already solved everything for a)

Yes, but you need to understand what the essential difference is between a and b.

 

[fabianprewett]

Ok well I know what the difference is, I just don't know what this implies or how this affects the final volume differently.

 

[haruspex]

I know what the difference is

So you can answer my question in post #8?

 

[Chestermiller]

For part b, draw a free body diagram of the piston and sand. Let $P_g (t)$ be the pressure of the gas on the lower face of the piston at time t (which may be different from the average pressure of the gas in the cylinder during this rapid irreversible compression). Assume that the piston and cylinder are in a vacuum, so that there is no pressure force acting downward on the top of the piston and sand. What does Newton's 2nd law give you for the force balance on the combination of piston and sand?