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Q.4 A Piston cylinder arrangement contains 0.05 m3 of a gas initially at 200 kPa. At this state a linear spring which has a spring constant of 150 kN/m is just touching the piston but exerting no force on it. Now heat is transferred to the gas causing the piston to rise and to compares the spring until the volume inside the cylinder doubles. If the cross section area of piston is 0.25 m2. Find (i)The final pressure inside the cylinder (ii) Work done by the gas?
Verified Answer
Q.4 A Piston cylinder arrangement contains 0.05 m3 of a gas initially ...
Given : Volume is 0.05 m�, k is 150 kN/m and area is 0.25 m� and pressure innitial is 200 kPa
To find: (a) the final pressure inside the cylinder,
(b) the total work done by the gas, and
(c) the fraction of this work against the spring to compress it.
Solution:
- (a) Now we have provided with volume, area and pressure, so the final pressure will be:
- Let the final pressure be P(f) and the innitial pressure be P(1).
- Now P(f) = P(1) + P(spring)
- But P(spring) = F(spring) / A = kx / A = k(ΔV/A) / A
k � ( (0.1 - 0.05) m� / 0.25 m� ) / 0.25 m�
50 kN/m � 0.2 m / 0.25 m�
30 kN / 0.25 m�
120 kPa
- So, P(f) = P(1) + P(spring)
- P(f) = 200 kPa + 120 kPa
- P(f) = 320 kPa
- (b). Now for total work done, let work done by piston be W1 and work done by spring be W2 and total work done be W.
- So, W1 + W2 = W
P(1)(V2 - V1) + k/2 x (x2� - x1�)
200 x (0.1 - 0.05) x (1 kJ/ 1 k Pa m�) + 1/2 x 150 x (0.2� - 0�) x (1 kJ/ 1 k Pa m�)
- After calculation, we get:
- W = 10 kJ + 3 kJ = 13 kJ
- (c). For work against the spring, let it be W(s). So:
W(s) = W(2) = k/2 x (x2� - x1�)
W(s) = 1/2 x 150 x (0.2� - 0�) x (1 kJ/ 1 k Pa m�)
W(s) = 3 kJ
Answer:
So, the final pressure inside the cylinder is 320 kPa, the total work done by the gas is 13 kJ and the fraction of this work against the spring to compress it is 3 kJ.
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Most Upvoted Answer
Q.4 A Piston cylinder arrangement contains 0.05 m3 of a gas initially ...
Given:
Initial volume of gas (V1) = 0.05 m^3
Initial pressure of gas (P1) = 200 kPa
Spring constant (k) = 150 kN/m
Cross-sectional area of piston (A) = 0.25 m^2
Final volume of gas (V2) = 2 * V1 = 2 * 0.05 m^3 = 0.1 m^3
(i) Final Pressure inside the Cylinder:
As the volume of gas doubles, the pressure inside the cylinder decreases.
According to Boyle's Law, the product of pressure and volume remains constant for a given amount of gas at constant temperature.
So, P1 * V1 = P2 * V2
200 kPa * 0.05 m^3 = P2 * 0.1 m^3
P2 = (200 kPa * 0.05 m^3) / 0.1 m^3
P2 = 100 kPa
Therefore, the final pressure inside the cylinder is 100 kPa.
(ii) Work Done by the Gas:
The work done by the gas can be calculated using the formula:
Work = Pressure * Change in Volume
In this case, the gas expands, so the work done by the gas is positive.
Change in Volume = Final Volume - Initial Volume
Change in Volume = V2 - V1
Change in Volume = 0.1 m^3 - 0.05 m^3
Change in Volume = 0.05 m^3
Work = Pressure * Change in Volume
Work = 100 kPa * 0.05 m^3
Work = 5000 J
Therefore, the work done by the gas is 5000 J.
Summary:
(i) The final pressure inside the cylinder is 100 kPa.
(ii) The work done by the gas is 5000 J.
Initial volume of gas (V1) = 0.05 m^3
Initial pressure of gas (P1) = 200 kPa
Spring constant (k) = 150 kN/m
Cross-sectional area of piston (A) = 0.25 m^2
Final volume of gas (V2) = 2 * V1 = 2 * 0.05 m^3 = 0.1 m^3
(i) Final Pressure inside the Cylinder:
As the volume of gas doubles, the pressure inside the cylinder decreases.
According to Boyle's Law, the product of pressure and volume remains constant for a given amount of gas at constant temperature.
So, P1 * V1 = P2 * V2
200 kPa * 0.05 m^3 = P2 * 0.1 m^3
P2 = (200 kPa * 0.05 m^3) / 0.1 m^3
P2 = 100 kPa
Therefore, the final pressure inside the cylinder is 100 kPa.
(ii) Work Done by the Gas:
The work done by the gas can be calculated using the formula:
Work = Pressure * Change in Volume
In this case, the gas expands, so the work done by the gas is positive.
Change in Volume = Final Volume - Initial Volume
Change in Volume = V2 - V1
Change in Volume = 0.1 m^3 - 0.05 m^3
Change in Volume = 0.05 m^3
Work = Pressure * Change in Volume
Work = 100 kPa * 0.05 m^3
Work = 5000 J
Therefore, the work done by the gas is 5000 J.
Summary:
(i) The final pressure inside the cylinder is 100 kPa.
(ii) The work done by the gas is 5000 J.
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Q.4 A Piston cylinder arrangement contains 0.05 m3 of a gas initially at 200 kPa. At this state a linear spring which has a spring constant of 150 kN/m is just touching the piston but exerting no force on it. Now heat is transferred to the gas causing the piston to rise and to compares the spring until the volume inside the cylinder doubles. If the cross section area of piston is 0.25 m2. Find (i)The final pressure inside the cylinder (ii) Work done by the gas?
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Q.4 A Piston cylinder arrangement contains 0.05 m3 of a gas initially at 200 kPa. At this state a linear spring which has a spring constant of 150 kN/m is just touching the piston but exerting no force on it. Now heat is transferred to the gas causing the piston to rise and to compares the spring until the volume inside the cylinder doubles. If the cross section area of piston is 0.25 m2. Find (i)The final pressure inside the cylinder (ii) Work done by the gas? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Q.4 A Piston cylinder arrangement contains 0.05 m3 of a gas initially at 200 kPa. At this state a linear spring which has a spring constant of 150 kN/m is just touching the piston but exerting no force on it. Now heat is transferred to the gas causing the piston to rise and to compares the spring until the volume inside the cylinder doubles. If the cross section area of piston is 0.25 m2. Find (i)The final pressure inside the cylinder (ii) Work done by the gas? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Q.4 A Piston cylinder arrangement contains 0.05 m3 of a gas initially at 200 kPa. At this state a linear spring which has a spring constant of 150 kN/m is just touching the piston but exerting no force on it. Now heat is transferred to the gas causing the piston to rise and to compares the spring until the volume inside the cylinder doubles. If the cross section area of piston is 0.25 m2. Find (i)The final pressure inside the cylinder (ii) Work done by the gas?.
Q.4 A Piston cylinder arrangement contains 0.05 m3 of a gas initially at 200 kPa. At this state a linear spring which has a spring constant of 150 kN/m is just touching the piston but exerting no force on it. Now heat is transferred to the gas causing the piston to rise and to compares the spring until the volume inside the cylinder doubles. If the cross section area of piston is 0.25 m2. Find (i)The final pressure inside the cylinder (ii) Work done by the gas? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Q.4 A Piston cylinder arrangement contains 0.05 m3 of a gas initially at 200 kPa. At this state a linear spring which has a spring constant of 150 kN/m is just touching the piston but exerting no force on it. Now heat is transferred to the gas causing the piston to rise and to compares the spring until the volume inside the cylinder doubles. If the cross section area of piston is 0.25 m2. Find (i)The final pressure inside the cylinder (ii) Work done by the gas? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Q.4 A Piston cylinder arrangement contains 0.05 m3 of a gas initially at 200 kPa. At this state a linear spring which has a spring constant of 150 kN/m is just touching the piston but exerting no force on it. Now heat is transferred to the gas causing the piston to rise and to compares the spring until the volume inside the cylinder doubles. If the cross section area of piston is 0.25 m2. Find (i)The final pressure inside the cylinder (ii) Work done by the gas?.
Solutions for Q.4 A Piston cylinder arrangement contains 0.05 m3 of a gas initially at 200 kPa. At this state a linear spring which has a spring constant of 150 kN/m is just touching the piston but exerting no force on it. Now heat is transferred to the gas causing the piston to rise and to compares the spring until the volume inside the cylinder doubles. If the cross section area of piston is 0.25 m2. Find (i)The final pressure inside the cylinder (ii) Work done by the gas? in English & in Hindi are available as part of our courses for GATE.
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