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 ?V 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?

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Answered by

Payal Gupta | Contributor-Level 10

4 months ago

This is a short answer type question as classified in NCERT Exemplar

For adiabatic change process we know

P₁V₁^y= P₂V₂^y

P (V+ $? V$ )^y = (P+ $? P$ )V^y

PV^y (1+ $\frac{? V}{V}$ ) ^y=p (1+ $\frac{? P}{P}$ )V^y

PV^y (1+ $\frac{? V}{V}$ ) $\approx$ PV^y (1+ $\frac{? V}{V}$ )

Y $\frac{? V}{V} = \frac{? P}{P}$

dV= $\frac{1}{Y} \frac{V}{p} d P$

hence work done increasing the pressure from P₁ to P₂

W= $\int_{p 1}^{p 2} p d V = \int_{p 2}^{p 1} p \frac{1}{Y} \frac{V}{p} d p$

= $\frac{V}{Y} (P_{2} - P_{1})$

W= $\frac{P_{2} - P_{1}}{Y} V$

This is a short answer type question as classified in NCERT ExemplarFor adiabatic change process we knowP1V1y= P2V2yP (V+ <math> <mo>? </mo> <mi>V</mi> </math> )y = (P+ <math> <mo>? </mo> <mi>P</mi> </math> )VyPVy (1+ <math> <mfrac> <mrow> <mrow> <mo>? </mo> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> </mfrac> </math> ) y=p (1+ <math> <mfrac> <mrow> <mrow> <mo>? </mo> <mi>P</mi> </mrow> </mrow> <mrow> <mrow> <mi>P</mi> </mrow> </mrow> </mfrac> </math> )VyPVy (1+ <math> <mfrac> <mrow> <mrow> <mo>? </mo> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> </mfrac> </math> ) <math> <mo>≈</mo> </math> PVy (1+ <math> <mfrac> <mrow> <mrow> <mo>? </mo> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> </mfrac> </math> )Y <math> <mfrac> <mrow> <mrow> <mo>? </mo> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>? </mo> <mi>P</mi> </mrow> </mrow> <mrow> <mrow> <mi>P</mi> </mrow> </mrow> </mfrac> </math> dV= <math> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mi>Y</mi> </mrow> </mrow> </mfrac> <mfrac> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mi>p</mi> </mrow> </mrow> </mfrac> <mi>d</mi> <mi>P</mi> </math> hence work done increasing the pressure from P1 to P2W= <math> <msubsup> <mo>∫</mo> <mrow> <mi>p</mi> <mn>1</mn> </mrow> <mrow> <mi>p</mi> <mn>2</mn> </mrow> </msubsup> <mrow> <mrow> <mi>p</mi> <mi>d</mi> <mi>V</mi> </mrow> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow> <mi>p</mi> <mn>2</mn> </mrow> <mrow> <mi>p</mi> <mn>1</mn> </mrow> </msubsup> <mrow> <mrow> <mi>p</mi> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mi>Y</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mfrac> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mi>p</mi> </mrow> </mrow> </mfrac> <mi>d</mi> <mi>p</mi> </math> = <math> <mfrac> <mrow> <mrow> <mi>V</mi> </mrow> </mrow> <mrow> <mrow> <mi>Y</mi> </mrow> </mrow> </mfrac> <mfenced separators="|"> <mrow> <mrow> <msub> <mrow> <mrow> <mi>P</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> <mo>-</mo> <msub> <mrow> <mrow> <mi>P</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfenced> </math> W= <math> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>P</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> <mo>-</mo> <msub> <mrow> <mrow> <mi>P</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <mi>Y</mi> </mrow> </mrow> </mfrac> <mi>V</mi> </math>

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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 ?V 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?

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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 ?V 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?

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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 ?V 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?