For diminishing isothermally n times the volume of gas in a cylinder with piston, over this piston is putted a load of mass m. What load should be added such that the volume of gas decreases isothermally k times more?
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<h3> Statement </h3>
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$5.5.3$
For diminishing isothermally $n$ times the volume of gas in a cylinder with piston, over this piston is putted a load of mass $m$. What load should be added such that the volume of gas decreases isothermally $k$ times more?
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For problem 5.5.3
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<h3>Solution</h3>
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Applying Newton's Second Law...
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Initially, considering a massless piston
$$P_0 = P_a \\;(1)$$
When load of mass $m$ is putted over
$$PS=mg+P_aS \\;(2)$$
After the addition of the another load,
$$P'S=(m+\Delta m)g+P_aS \\;(3)$$
From Boyle-Mariotte Law
$$P_0V_0 = P\frac{V_0}{n}$$
$$P_0 = \frac{P}{n} \\;(4)$$
Substituting $(2)$ into $(3)$, according to $(1)$ and separating $P_a$
$$P_a = \frac{mg}{S(n-1)} \\;(5)$$
Applying Boyle-Mariotte Law again
$$P\frac{V_0}{n}=P'\frac{V_0}{nk}$$
$$P=\frac{P'}{k} \\;(6)$$
Substituting $(2)$ and $(3)$ into $(6)$ and developing algebraically
$$\left(\frac{mg}{S}+P_a\right)(k-1)=\frac{\Delta mg}{S} \\;(7)$$
Putting $(5)$ into $(7)$ and separating $\Delta m$
$$\boxed{\Delta m=m\frac{(k-1)n}{(n-1)}}$$
This problem can also be solved considering the mass of piston.
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<p style="text-align: right; font-style: italic; font-size: 14;">
BSc. Luis Daniel Fernández Quintana<br>
Physics Department (FCNE)<br>
Universidad de Oriente, Cuba<br>
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