ParthKohli
  • ParthKohli
Try this sweet physics question. I'm not bragging that I was able to solve this, but I am.
Mathematics
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SOLVED
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jamiebookeater
  • jamiebookeater
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
ParthKohli
  • ParthKohli
A certain mass of an ideal gas is enclosed in a cylinder of volume \(V_0\) fitted with a smooth heavy piston of mass \(m\) and cross-sectional area \(A\). The piston is displaced through a small distance downwards so as to compress the gas isothermally, and then it is let go. Take the atmospheric pressure as \(P\). Find the time period of the resultant oscillations.
alekos
  • alekos
I've never seen a physics question like this before!! Let me think about it
Kainui
  • Kainui
This sounds like a perpetual motion device lol

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More answers

ParthKohli
  • ParthKohli
That is because it is in ideal conditions. But if we extract energy, it should stop its motion, so it isn't a perpetual motion in that sense (you cannot extract infinite energy).
IrishBoy123
  • IrishBoy123
start with the ideal gas law \(pV = nRT = const \) as isothermal do \(d(pV) = 0 = pdV + V dp \implies dp = -\frac{p dV}{V}\) from there, for ***small*** displacement x, the restoring "linear" force for shm is \(dp A\), so \( |dp A| = - m \ddot x\). etc
Michele_Laino
  • Michele_Laino
I think that the system can be modeled with the subsequent 2-nd order ODE \[\Large M\frac{{{d^2}\left( {PS} \right)}}{{d{t^2}}} = - \left( {P - {P_0} - Mg} \right)S\] where \(S\) is the surface of the piston, \(M\) is its mass, and \(P_0\) is the atmospheric preesure
Michele_Laino
  • Michele_Laino
oops..pressure*
alekos
  • alekos
I get f''[x(t)] + xPA^2/mV = 0 where P = p + mg/A (p = atmospheric pressure)
alekos
  • alekos
looks similar to Michele's
alekos
  • alekos
and for the period i get T = 2πsqrt(mV/PA^2)
alekos
  • alekos
how does that look?
Michele_Laino
  • Michele_Laino
oops.. I have made a typo, here is the right equation |dw:1449332945003:dw| \[\Large \frac{{{d^2}z}}{{d{t^2}}} = \frac{{ - \left( {P - {P_0} - Mg} \right)S}}{M},\quad P = \frac{{nRT}}{{Sz}}\]
anonymous
  • anonymous
All I have to say is that being able to solve a question like this is my goal. Good job!
Michele_Laino
  • Michele_Laino
more precisely, we can write this: \[\Large P = \frac{{nRT}}{{Sz}},\quad {P_0} + Mg = \frac{{nRT}}{{S{z_0}}}\] and for little oscillations, namely \[\huge z \approx {z_0}\] we have: \[\huge \ddot z + \frac{{nRT}}{{Mz_0^2}}\left( {z - {z_0}} \right) = 0\] therefore, the period of little oscillations is: \[\huge T = 2\pi \left( {\frac{{{V_0}}}{S}} \right)\sqrt {\frac{M}{{nRT}}} \]
alekos
  • alekos
so @parthkohli what's the answer
alekos
  • alekos
he's disappeared and left us holding the baby!
ParthKohli
  • ParthKohli
Hey I'm back!
ParthKohli
  • ParthKohli
Basically you're not given \(n, T\) but I'm sure you can use the equation of state.
ParthKohli
  • ParthKohli
@alekos looks like you're right!
alekos
  • alekos
Awesome! thanks @parthkohli great question!
Michele_Laino
  • Michele_Laino
we can make this substitution: \[\huge nRT = {P_0}{V_0}\] so, we get: \[\huge T = 2\pi \sqrt {\frac{{M{V_0}}}{{{P_0}{S^2}}}} \]
ParthKohli
  • ParthKohli
Substitute \(P = P_{atm} + mg/A\) for the actual final answer though.
ParthKohli
  • ParthKohli
Again, the substitution is there because you're not explicitly given the initial pressure of the gas.
Michele_Laino
  • Michele_Laino
please wait, I have made an error in such formula \(P_0\) is the atmospheric pressure, so here are the right formulas: \[\huge nRT = \left( {{P_0} + \frac{{Mg}}{S}} \right){V_0}\] and: \[\huge T = 2\pi \left( {\frac{{{V_0}}}{S}} \right)\sqrt {\frac{M}{{\left( {{P_0} + \frac{{Mg}}{S}} \right)V}}} \]
ParthKohli
  • ParthKohli
Yes, that's it.
Michele_Laino
  • Michele_Laino
:)
ParthKohli
  • ParthKohli
Well done guys, good work! I'll post more questions tomorrow.
Michele_Laino
  • Michele_Laino
ok! :)
alekos
  • alekos
good on ya @Michele
Michele_Laino
  • Michele_Laino
thanks!! :) @alekos
alekos
  • alekos
3 medals compared to my 2
Michele_Laino
  • Michele_Laino
I gave you my medal lol :)
Michele_Laino
  • Michele_Laino
sorry I gave it to @IrishBoy123
alekos
  • alekos
thanks mate!!
IrishBoy123
  • IrishBoy123
@alekos i have passed it on to you :p
alekos
  • alekos
thanks @IrishBoy123 Trying to get to 99

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