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Consider a cycle tyre being filled with...

Consider a cycle tyre being filled with air by a pump. Let V be the volume of the tyre (fixed) and at each stroke of the pump `DeltaV(ltltV)` of air is transferred to the tube adiabatically. What is the work done when the pressure in the tube is increased from `p_(1)` to `p_(2)` ?

Text Solution

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Let, volume is increased by `DeltaV` and pressure is increased by `Deltap` by an stroke.
For just before and after an stroke, we can write
`" "p_(1)V_(1)^(gamma)=p_(2)V_(2)^(gamma)`
`rArr" "p(V+DeltaV)^(gamma)=(p+Deltap)V^(gamma)" "(because" volume is fixed")` ltBrgt `rArr" "pV^(gamma)(1+(DeltaV)/(V))^(gamma)=p(1+(Deltap)/(p))V^(gamma)`
`rArr" "pV^(gamma)(1+gamma(DeltaV)/V)~~pV^(gamma)(1+(Deltap)/(p))" "(because Deltavltltv)`
`rArr" "gamma(DeltaV)/(V)=(Deltap)/(p)rArrDeltaV=(1)/(gamma)(V)/(p)Deltap`
`rArr" "dV=(1)/(gamma)(V)/(p)dp`
Hence, work done is increasing the pressure form `p_(1)` to `p_(2)`
`W=int_(p_(1))^(p_(2))pdV=int_(p_(1))^(p^(2))pxx(1)/(gamma)(V)/(p)dp`
`" "=(V)/(gamma)int_(p_(1))^(p_(2))dp=(V)/(gamma)(p_(2)-p_(1))`
`rArr" "W=((p_(2)-p_(1)))/(gamma)V`
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