Thermodynamics problem -- Pressure oscillations in a jar

AI Thread Summary
The discussion revolves around a thermodynamics problem involving pressure oscillations in a jar. Participants clarify the significance of the negative sign in the force equation, emphasizing that it indicates opposite directions, even when treating the variables as scalars. They explore the relationship between pressure and volume changes, specifically questioning the validity of certain equations and assumptions regarding small displacements. The conversation also addresses the Taylor series expansion for small values, leading to a clearer understanding of pressure changes in relation to volume displacement. Overall, the dialogue enhances comprehension of the dynamics at play in the thermodynamic system.
LagrangeEuler
Messages
711
Reaction score
22
I add a Figure with the problem and solution. I have difficulty with a solution to the given problem. Why ##F=-kx=Adp##, I do not understand minus sign because we are working with scalars not vectors. It is correct to say that
##\vec{F}=-kx\vec{i}##, but is not correct to say that ##F=-kx##. Can you explain to me this part?
 

Attachments

  • problem.png
    problem.png
    33.1 KB · Views: 158
Science news on Phys.org
LagrangeEuler said:
It is correct to say that ##\vec{F}=-kx\vec{i}##, but is not correct to say that ##F=-kx##. Can you explain to me this part?
I would prefer ##\ \vec F = - k\vec x\ ##. The minus sign indicates that the directions are opposite. For motion along a straight line one can dispose of the vector character, but the minus sign stays:$$
\vec F = - k\vec x \Leftrightarrow F \hat \imath = - k x\hat \imath\quad \Rightarrow \quad F = - k x $$
 
  • Like
Likes LagrangeEuler
Is it correct to write
(p_0+\frac{mg}{A})V_0^{\gamma}=p_2(V_0-Ax)^{\gamma}?
From that
p_2=\frac{(p_0+\frac{mg}{A})V_0^{\gamma}}{(V_0-Ax)^{\gamma}}
and
p_2=\frac{p_0+\frac{mg}{A}}{(1-\frac{Ax}{V_0})^{\gamma}}
Can we assume here that ##x## is very small so that
(1+x)^n \approx 1+nx?
From that
p_2=\frac{p_0+\frac{mg}{A}}{1-\frac{\gamma A x}{V_0}}
 
LagrangeEuler said:
Is it correct to write
(p_0+\frac{mg}{A})V_0^{\gamma}=p_2(V_0-Ax)^{\gamma}\ \ ?
Your book says ##\ x = dV/A\ ## so it chooses a coordinate system with upwards as positive x-direction.
So I would write ##V_2 = V_0 + Ax##. Other than that: yes.

One small thing: x can be many ##\mu##m, however. What you mean is ##\displaystyle {{Ax\over V}<< 1}\ ##.

From your last expression (sign corrected) : $$p_2 = \left (p_0+\frac{mg}{A}\right )\ \left (1 - \frac{\gamma A^2 x}{V_0}\right )\quad \Rightarrow \quad \Delta F = p_2A-p_1A=-\frac{\gamma A^2 }{V_0}\left (p_0+\frac{mg}{A}\right )\,x$$

##\ ##
 
  • Like
Likes LagrangeEuler and vanhees71
LagrangeEuler said:
Is it correct to write
(p_0+\frac{mg}{A})V_0^{\gamma}=p_2(V_0-Ax)^{\gamma}?
From that
p_2=\frac{(p_0+\frac{mg}{A})V_0^{\gamma}}{(V_0-Ax)^{\gamma}}
and
p_2=\frac{p_0+\frac{mg}{A}}{(1-\frac{Ax}{V_0})^{\gamma}}
Can we assume here that ##x## is very small so that
(1+x)^n \approx 1+nx?
From that
p_2=\frac{p_0+\frac{mg}{A}}{1-\frac{\gamma A x}{V_0}}
I like what you did here. But, now you can take it one step further by continuing the expansion for small x: $$p_2=(p_0+\frac{mg}{A})+(p_0+\frac{mg}{A})\frac{\gamma A x}{V_0}$$So, $$\Delta p=(p_0+\frac{mg}{A})\frac{\gamma A x}{V_0}$$So the net upward force on the ball is $$F=A\Delta p=(p_0+\frac{mg}{A})\frac{\gamma A^2 }{V_0}x$$where x is the downward displacement.
 
  • Like
Likes LagrangeEuler and vanhees71
BvU said:
Your book says ##\ x = dV/A\ ## so it chooses a coordinate system with upwards as positive x-direction.
So I would write ##V_2 = V_0 + Ax##. Other than that: yes.

One small thing: x can be many ##\mu##m, however. What you mean is ##\displaystyle {{Ax\over V}<< 1}\ ##.

From your last expression (sign corrected) : $$p_2 = \left (p_0+\frac{mg}{A}\right )\ \left (1 - \frac{\gamma A^2 x}{V_0}\right )\quad \Rightarrow \quad \Delta F = p_2A-p_1A=-\frac{\gamma A^2 }{V_0}\left (p_0+\frac{mg}{A}\right )\,x$$

##\ ##
Thanks a lot. Could you just explain me better part ##V_0-Ax## ##\rightarrow## ##V_0+Ax##? I have problem to understand why ##V_2=V_0+Ax## and relation of this and ##x=dV/A##.
 
The Taylor series expansion for ##\frac{1}{1-\frac{\gamma Ax}{V_0}}## is ##\left(1+\frac{\gamma Ax}{V_0}+...\right)##

To linear terms in x, this is $$\frac{1}{1-\frac{\gamma Ax}{V_0}}\approx \left(1+\frac{\gamma Ax}{V_0}\right)$$
 
LagrangeEuler said:
Thanks a lot. Could you just explain me better part ##V_0-Ax## ##\rightarrow## ##V_0+Ax##? I have problem to understand why ##V_2=V_0+Ax## and relation of this and ##x=dV/A##.
If the ball moves up by a distance ##x##, the volume increases by ##Ax\ ##, that's all :smile: !
 
  • Like
Likes vanhees71
Back
Top