Consider a cycle tyre being filled

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?


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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