Thermodynamics problem - weight and piston connected by string

[Saitama]

Homework Statement

An easily moveable, thermally insulated piston of negligible mass confines a sample of air of volume $V_0=8\,\, \text{litre}$, at a temperature of $T_0=300 K$, into a thermally insulated cylinder of cross section $A=1\,\,dm^2$. The cylinder is fixed to a horizontal table. The pressure of the confined air is the same as the atmospheric pressure $p_0=105 \,\,Pa$. A piece of thread is attached to the centre of the piston, it lies in the line of the axis of the cylinder and leads through a pulley. A weight of mass $m=25\,\, kg$ is attached to the other end of the thread as shown in the figure. Holding the weight, any part of the thread is straight, but loose.

The weight is released without any push. To an accuracy of 1% determine the

a) temperature of the air in the cylinder, when the weight is at its lowermost position;

b) the acceleration of the piston at the lowermost position of the weight;

c) the greatest speed of the piston during the process. (The mass of the thread and the pulley and friction are negligible.)

Homework Equations

The Attempt at a Solution

I am thinking of using conservation of energy but I am not sure which expansion process should I consider. If I consider adiabatic, I need the adiabatic constant which is not mentioned in the problem statement. I can't think of any other process which takes place in a thermally insulated cylinder.

Any help is appreciated. Thanks!

 

[NoPoke]

What simplifying assumptions could you make?

 

[DrClaude]

confines a sample of air

If I consider adiabatic, I need the adiabatic constant which is not mentioned in the problem statement.

Put these two quotes together.

 

[Saitama]

Put these two quotes together.

Sorry, I can't really think of anything.

 

[DrClaude]

Can you figure out what the adiabatic constant for air could be?

 

[Saitama]

Can you figure out what the adiabatic constant for air could be?

The question doesn't state the nature of gas (air) i.e is it monoatomic, diatomic or something else? I don't see how to find the adiabatic constant without this information.

 

[NoPoke]

What gases will you find in air?
What is their composition (monatomic , diatomic, etc)?
If one type is predominant then what?

 

[DrClaude]

The question doesn't state the nature of gas (air) i.e is it monoatomic, diatomic or something else? I don't see how to find the adiabatic constant without this information.

What is air mostly composed of?

 

[Saitama]

What is air mostly composed of?

Okay, I took it as diatomic so $\gamma=7/5$ (adiabatic constant). At the lowermost position of block, the kinetic energy of block is zero, so from conservation of energy:
$$mgx+p_{atm}Ax=\frac{p_0V_0-P'(V_0+Ax)}{\gamma-1}$$
and
$$P'=\frac{p_0V_0^{\gamma}}{(V_0+Ax)^{\gamma}}$$
where $p_{atm}$ is the pressure due to air outside and x is the displacement of block. Solving the above gives me a negative value for $x$. :(

 

[DrClaude]

I'm not sre where you are going with this. I would start by considering the final state, when the piston is at equilibrium again, in terms of the forces acting on it.

 

[Saitama]

I'm not sre where you are going with this. I would start by considering the final state, when the piston is at equilibrium again, in terms of the forces acting on it.

Why equilibrium? How does it justify that at equilibrium of piston, the block is at lowermost position? What's wrong with my equation for conservation of energy?

 

[DrClaude]

Why equilibrium? How does it justify that at equilibrium of piston, the block is at lowermost position? What's wrong with my equation for conservation of energy?

I'm sorry if I'm confusing you. I was neglecting the inertia of the block, which might not be a good approximation here.

Could you clear up your thinking in writing

$$mgx+p_{atm}Ax=\frac{p_0V_0-P'(V_0+Ax)}{\gamma-1}$$
and
$$P'=\frac{p_0V_0^{\gamma}}{(V_0+Ax)^{\gamma}}$$

 

[Saitama]

Could you clear up your thinking in writing

Sure! :)

I set x=0 and t=0 when the string tightens. Let the displacement of block when it is at lowermost position be $x$. Since there is no change in kinetic energy, the sum of work done by gravity on block, work done in expansion by confined and the work done by the air outside on the piston is zero i.e
$$-mgx+\frac{p_0V_0-P'(V_0+Ax)}{\gamma-1}-p_{atm}x=0$$
which is the same equation I wrote above, please let me know if anything is still unclear.

 

[voko]

Why is the work done by gravity negative?

And frankly, I am not convinced your middle term is correct. There are many ways to express work in adiabatic expansion, but I do not think this is one of them. Where does that come from?

 

[Saitama]

Why is the work done by gravity negative?

Oops, very sorry.

But I still don't get the answer. :(

And frankly, I am not convinced your middle term is correct. There are many ways to express work in adiabatic expansion, but I do not think this is one of them. Where does that come from?

I used the following formula:
$$W=\frac{P_iV_i-P_fV_f}{\gamma-1}$$
http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html

 

[voko]

I used the following formula:
$$W=\frac{P_iV_i-P_fV_f}{\gamma-1}$$
http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html

I do not see such a formula there.

 

[Saitama]

I do not see such a formula there.

Okay let me explain. The formula mentioned on that page is:
$$W=\frac{K(V_f^{1-\gamma}-V_i^{1-\gamma})}{1-\gamma}$$
Factoring out -1 from the denominator:
$$W=\frac{K(V_i^{1-\gamma}-V_f^{1-\gamma})}{\gamma-1}$$
Also, $K=PV^{\gamma}$, hence, $KV_f^{1-\gamma}=P_fV_f\cdot V_f^{1-\gamma}=P_fV_f$, similarly, $KV_i^{1-\gamma}=P_iV_i$. I hope everything is clear now. :)

 

[voko]

Clear indeed.

So what equation do you obtain in the end?

 

[Saitama]

Clear indeed.

So what equation do you obtain in the end?

This is what I have:
$$mgx+\frac{p_0V_0-P'(V_0+Ax)}{\gamma-1}-p_{atm}x=0$$
where
$$P'=\frac{p_0V_0^{\gamma}}{(V_0+Ax)^{\gamma}}$$
I simplify the middle term in the energy equation:
$$\begin{align*}
\frac{p_0V_0-P'(V_0+Ax)}{\gamma-1} & = \frac{1}{\gamma-1}\left(p_0V_0-\frac{p_0V_0^{\gamma}}{(V_0+Ax)^{\gamma}}(V_0+Ax)\right)\\ & = \frac{p_0V_0}{\gamma-1}\left(1-\left(1+\frac{Ax}{V_0}\right)^{1-\gamma}\right)\\
\end{align*}$$
So the energy equation becomes:
$$mgx+\frac{p_0V_0}{\gamma-1}\left(1-\left(1+\frac{Ax}{V_0}\right)^{1-\gamma}\right)=p_{atm}Ax$$
Plugging this in Wolfram Alpha doesn't give me the answer. :(

http://www.wolframalpha.com/input/?i=25*9.8*x+(105*8)/(2/5)*(1-(1+10^(-2)x/8)^(-2/5))=10^5*10^(-2)*x

I take $P_{atm}\approx 10^5 \,\,Pa$.

 

[voko]

There are a couple of errors. In one case, you use 105 for pressure. More seriously, you use 8 for the volume - is it really 8 cubic metres?

 

[Saitama]

There are a couple of errors. In one case, you use 105 for pressure. More seriously, you use 8 for the volume - is it really 8 cubic metres?

Ah, I am very sorry, I made the correction but I still don't get the right answer.

I replaced 8 litres with 0.008 $m^3$. http://www.wolframalpha.com/input/?...*(1-(1+10^(-2)x/0.008)^(-2/5))=10^5*10^(-2)*x

Is there something wrong with the pressure? It is given in the problem statement that $p_0$ is $105\,\,Pa$.

 

[voko]

The problem states that the initial pressure inside the cylinder is equal to atmospheric. So you have to use the same number for both at the very least. And 105 Pa is probably an error, where 10^5 Pa was meant.

 

[Saitama]

The problem states that the initial pressure inside the cylinder is equal to atmospheric. So you have to use the same number for both at the very least. And 105 Pa is probably an error, where 10^5 Pa was meant.

Sorry for doing so many mistakes, and the typo in the problem statement is due to me, I should have checked again.

I will be attempting the other two parts tomorrow, I need to leave now. Thanks for all the help voko!

 

[Chestermiller]

This is basically going to be analogous to a SHM problem, except that the spring (in this case, the gas in the cylinder) is nonlinear. If T is the tension in the string, then a force balance on the piston gives:

T = (P₀-P)A, where P is the pressure in the cylinder, and P₀=P_atm=10⁵ Pa.

A force balance on the mass gives:

$$mg-T=m\frac{d^2x}{dt^2}$$

The gas is expanded and compressed adiabatically, so

PV^γ=P₀V₀^γ

or, equivalently,
$$P=P_0\left(\frac{V_0}{V_0+Ax}\right)^γ$$
Combining the previous equations gives:
$$m\frac{d^2x}{dt^2}=mg-AP_0(1-\left(\frac{V_0}{V_0+Ax}\right)^γ)$$
If we multiply this equation by v = dx/dt and integrate from 0 to x, we obtain:
$$m\frac{v^2}{2}=mgx-AP_0x+\frac{P_0V_0}{γ-1}\left(1-\left(\frac{V_0}{V_0+Ax}\right)^{γ-1}\right)$$

This confirms Panov-Arora's result for the case in which v = 0. This corresponds to the lowest point reached by the mass. The point of maximum velocity is where the acceleration= 0 (this is also the equilibrium point). This is where:
$$AP_0(1-\left(\frac{V_0}{V_0+Ax}\right)^γ)=mg$$

Chet

 

[Saitama]

$$m\frac{d^2x}{dt^2}=mg-AP_0(1-\left(\frac{V_0}{V_0+Ax}\right)^γ)$$

I got the exact same equation when I tried the problem with Newton's second law but I fail to see how this is SHM. Please help.

For the b) part, I solved the problem using Newton laws. And the equation I used is similar to the Chestermiller.

For the c) part, I used energy conservation. But first, I found $x$ from the Newton's second law. The velocity is maximum at the equilibrium point. From Newton's second law:
$$m\frac{d^2x}{dt^2}=mg-Ap_0\left(1-\left(1+\frac{Ax}{V_0}\right)^{-\gamma}\right)=0$$
Solving for $x$ gives $x=0.177847\,\,m$.
From energy conservation,
$$mgx+\frac{p_0V_0}{\gamma-1}\left(1-\left(1+\frac{Ax}{V_0}\right)^{1-\gamma}\right)-p_{atm}Ax=\frac{1}{2}mv^2$$
But solving for $v$ doesn't give me the right answer.

http://www.wolframalpha.com/input/?...2/5))-10^5*10^(-2)*x=1/2+*+25*y^2,+x=0.177847

The given answer is $0.4\,\,m/s$ but I get $1.2\,\, m/s$. :(

 

[voko]

I got the exact same equation when I tried the problem with Newton's second law but I fail to see how this is SHM. Please help.

It is not SHM per se. But for small $x$, it is. Linearise the right-most term.

I am not sure why what you do for (c) is wrong. It looks correct but I will have another look later on. Perhaps some numeric errors again?

 

[Chestermiller]

I got the exact same equation when I tried the problem with Newton's second law but I fail to see how this is SHM. Please help.

I didn't say it was SHM. I said it was analogous to SHM. As Voko implied, if you linearize about the equilibrium location x = 0.178 m, you get the differential equation for SHM. From this, you can actually determine the spring constant k for the cylinder/piston "spring." I recommend that you try it and see.

I confirmed your value for the equilibrium length, and, when I substituted this into your equation for the velocity, I got 1.27 m/s.

Incidentally, I assume there was a mistake with regard to the mass m in the problem statement. Apparently, m = 25 gm, not 25 kg.

Chet

 

[Saitama]

Incidentally, I assume there was a mistake with regard to the mass m in the problem statement. Apparently, m = 25 gm, not 25 kg.

Nope, I rechecked the question, the mass is given to be 25 kg.

http://www.komal.hu/verseny/feladat.cgi?a=feladat&f=P4601&l=en

 

[Chestermiller]

Nope, I rechecked the question, the mass is given to be 25 kg.

http://www.komal.hu/verseny/feladat.cgi?a=feladat&f=P4601&l=en

Oops. You're right. I made compensating errors with units to get the correct equilibrium location and the velocity. So my result of 1.27 m/s still stands.

Chet

 

[Saitama]

Oops. You're right. I made compensating errors with units to get the correct equilibrium location. I'll redo my calculations for the velocity.

Chet

I have used the same values given in the problem, what is wrong with my calculations?

 

[Chestermiller]

I have used the same values given in the problem, what is wrong with my calculations?

As I said in my previous edited posting, I redid my calculations, and obtained 0.178 m for the equilibrium position and 1.27 m/s for the velocity. This velocity agrees with your value.

Chet

 

[Saitama]

As I said in my previous edited posting, I redid my calculations, and obtained 0.178 m for the equilibrium position and 1.27 m/s for the velocity. This velocity agrees with your value.

Chet

Ah, then the given answer must be wrong.

Thanks for the help voko, Chestermiller and DrClaude!