Question

The figure below shows a gas contained in a vertical piston– cylinder assembly, where D = 10 cm. A vertical shaft whose cross-sectional area is 0.8 cm2 is attached to the top of the piston. The total mass of the piston and shaft is 25 kg. While the gas is slowly heated, the internal energy of the gas increases by 0.1 kJ, the potential energy of the piston–shaft combination increases by 0.2 kJ, and a force of F = 1668 N is exerted on the shaft as shown in the figure. The piston and cylinder are poor conductors, and friction between them is negligible.The local atmospheric pressure is 1 bar and g = 9.81 m/s2.

Determine the work done by the shaft, in kJ.

Answer 1

The work done by the force on the moving shaft is the product of the

height moved by the shaft and the magnitude of the force.

Correct response:

• The work done by the shaft is approximately 1,367.6 J

Methods used to calculate work done by a force

Given parameters are;

Diameter of the piston, D = 10 cm

Mass of the piston and shaft = 25 kg

The increase in the internal energy of the gas = 0.1 kJ

Potential energy of the piston-shaft combination = 0.2 kJ

Force of applied = 1,668 N

Cross-sectional area of the shaft = 0.8 cm²

Local atmospheric pressure = 1 bar

Required:

The work done by the shaft

Solution:

Potential energy, P.E. = ρ·g·h

$h = \mathbf{\dfrac{P.E.}{\rho \cdot g}}$

Which gives;

$Displacement \ of \ piston \ and \ shaft, \, h = \mathbf{\dfrac{0.2 \, kJ}{25 \, kg \times 9.81 \, m/s^2}} \approx0.82 \, m$

Work done = Force × Distance

Therefore;

Work done, W ≈ 1,668 N × 0.82 m = 1,367.76 J

• Work done by the shaft, W ≈ 1,367.76 J

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