Physics Homework #171 Heat And Thermodynamics In Physics

This is an old physics olympiad problem, I think. The answer hinges on the spheres expanding due to heating. Sphere A raises its center of mass some and sphere B lowers its center of mass some. By conservation of energy sphere A is thus slightly colder, since more of its energy went into its gravitational potential.

It's a bit of a silly problem since the effect is extremely small. We can see this because from common experience if you heat a metal sphere $10 C$, the change in radius is pretty small - so small you probably won't notice without measuring it or else having something with a different expansion coefficient wrapped around the sphere. Meanwhile, if you drop a fist-sized a metal sphere by $1cm$, a distance much larger than such a sphere would expand with a $10 C$ change, the temperature change is much, much smaller than $10 C$. It's smaller than you can even notice, really. So the gravitational potential change is very small compared to the heat, and the difference in temperatures is minute.

Heated Piston

A cylinder of cross-sectional area 0.0314m^2, filled with argon (a monatomic ideal gas), is sealed with a piston of mass M=25kg that is free to move up and down. The piston is initially at a height of 30cm, and the gas is at 300K. The cylinder is heated from below. After a few minutes, 171J of heat has been added to the system, and the piston has moved upward by 2cm. See figure 1. This problem has multiple ways to calculate the same quantities. This can be a useful check if you are unsure of your answer.

(a) Draw a free body diagram to show all the forces acting on the piston (include the force of the atmosphere pushing down on the piston from outside). (b) What is the initial pressure of the gas in the cylinder? Is this pressure constant? (hint: think about the forces on the piston) (c) How much work is done by the gas in moving the piston 2cm upward? (note: there are two ways to calculate this based on the information you have) (d) Find the change in internal energy (using the first law). (e) Find the change in temperature of the gas using the ideal gas law, and the number of moles of gas present, then find the change in internal energy using !U = nCV !T for an ideal diatomic gas? The numbers from both methods should agree (there may be a small discrepancy due to rounding). If not, review previous steps to find your error

My work so far:

b)

I used the following equations,
F=pA (pressure x area of piston)
F=ma for the force of gravity acting on the piston
I made the assumption that the pressure acting down on the piston was from the force of gravity, and so made these two equations equal to one another.
pA=ma
p=(ma)/A
p=[(25kg)(9.8m/s^2)]/(.0314m^2)
p=7803kg/ms^2
I know that these are the correct units for pressure, and I also felt that the pressure must be fairly large in order to support such a heavy mass.

c)W=pΔV (ΔV=change in volume of cylinder)
ΔV is unknown so I had to solve for it
The volume of a cylinder = πr^2h = area of base x height = Ah
The change in volume here is a result of a change in height of the piston
so, ΔV = AΔh
ΔV= (.0314m^2)(.002m)
ΔV= .0000628m^3
And then I plugged it into W=pΔV
W=(7803kg/ms^2)(.0000628m^3)
W=(0.49kgm^2/s^2)
W=0.49J
Again, I knew I had the right units which was good.

D) I used the equation, ΔU = Q-W, where ΔU is change in internal energy
I plugged in the given value for Q (171J) and the solved value of W (0.49J)
ΔU = 171-0.49= 170.51 J

E) I used the equation for the ideal gas law, PV=nRT
I used the values of the cylinder in it's initial state in order to solve for n.
n=PV/RT
n=[(7803)(.0314)(.032)]/[(8.314)(300)]
n=.00395 moles of gas
I definitely felt that this number was pretty low but I decided to try to find temperature anyway. T=PV/nR
T=[(7803)(.0314)(.032)]/[(.00295)(8.314)]
T=320 K

I knew this answer fit with what my estimation should be, as heat has been added to the system this final temperature should be higher than the initial temperature of 300K. However, the low value I found for moles of gas is throwing me off. I was hoping that somebody could check my work for error, or to work through it and compare the answers.

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