# 英国代考|6CCM331A Special Relativity And Electromagnetism狭义相对论与电磁学

Let the initial pressures in the upper and lower compartment be $p_{1}$ and $p_{2}$. Let the final pressures be $p_{1}^{\prime}$ and $p_{2}^{\prime}$. In order for the system to be in mechanical equilibrium, the pressure differences must balance the pressure due to the weight of the piston (and the weight of the ball in the first case).
\begin{aligned} &p_{2}=p_{1}+\frac{(m+M) g}{A}, \ &p_{2}^{\prime}=p_{1}^{\prime}+\frac{M g}{A} . \end{aligned}

Next, we know that gases must have common initial and final temperatures. Then, the ratio of moles in the two compartments are given by the ideal gas law as
$$\frac{n_{1}}{n_{2}}=\frac{p_{1} V_{1}}{p_{2} V_{2}}=\frac{p_{1}^{\prime} V_{1}^{\prime}}{p_{2}^{\prime} V_{2}^{\prime}}$$
Substituting $\frac{V_{1}}{V_{2}}=3, \frac{V_{1}^{\prime}}{V_{2}^{\prime}}=2$ and the expressions for $p_{2}$ and $p_{2}^{\prime}$ in terms of $p_{1}$ and $p_{1}^{\prime}$,
$$p_{1}^{\prime}=\frac{3 p_{1} M g}{2(m+M) g-p_{1} A}$$

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