Consider a cycle tyre being filled
Question:

Consider a cycle tyre being filled with air by a pump. Let V be the volume of the tyre and at each stroke of the pump ∆V of air is transferred to the tube adiabatically. What is the work done when the pressure in the tube is increased from P1 to P2?

Solution:

Following is the equation before and after the stroke:

$P_{1} V_{1}^{\gamma}=P_{2} V_{2}^{\gamma}$

$P(V+\Delta V)^{\gamma}=(P+\Delta P) V^{\gamma} \Rightarrow P V^{\gamma}\left(1+\frac{\Delta V}{V}\right)^{\gamma}=P\left(1+\frac{\Delta P}{P}\right) V^{\gamma}$

$P V^{\gamma}\left(1+\gamma \frac{\Delta V}{V}\right) \approx P V^{\gamma}\left(1+\frac{\Delta P}{P}\right)$

$\gamma \frac{\Delta V}{V}=\frac{\Delta P}{P}$

Therefore, work done is given as

$W=\frac{\left(P_{2}-P_{1}\right) V}{\gamma}$

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