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A gas is confined in a 1.25(ft) diameter cylinder by a piston, on which rests a weight. The mass of
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Answer
The force in (lb f) exerted on the gas by the atmosphere, the piston, and the weight,
assuming no friction between the piston and cylinder is 8080 lb f. The pressure of
the gas in (psia) is 6594.1 psia. The work done by the gas in (ft)(lb f) is 1957.95 ft lb
f. The change in potential energy of the piston and weight is 14147.19 ft lb f.
Explanation
To determine the force exerted on the gas within the cylinder, we can follow a
structured approach.
1 Calculate the area of the piston. The diameter of the piston is given as 1.25 feet,
so the radius is half of that, which is 0.625 feet. The area (A) can be calculated
using the formula for the area of a circle:
A = πr2 .
Substituting the radius, we find:
A = π(0.625)2 ≈ 1.227 f t2 .
2 Calculate the force exerted by atmospheric pressure. The atmospheric pressure
is given as 30.12 inches of mercury (in Hg). To convert this to pounds per square
foot (lb f/ft²), we use the conversion factor:
1 in Hg = 0.4912 lb f/in².
Thus, the atmospheric pressure in lb f/ft² is:
144 in2
patm = 30.12 in Hg × 0.4912 lb f /in2 /in Hg × ≈ 38.1 lb f .
1 f t2
3 Calculate the weight of the piston and the weight resting on it. The total weight
(W) is given as 250 lb m. To convert this to lb f, we multiply by the acceleration
due to gravity (32.169 ft/s²):
W = 250 lb m × 32.169 f t/s2 ≈ 8042.25 lb f .
4 Combine the forces to find the total force (F) exerted on the gas:
Gauth F = patm ⋅ A + W ≈ 38.1 lb f + 8042.25 lb f ≈ 8080 lb f .
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Gauth AI Pro Next, we calculate the pressure of the gas:
PDF Helper 05. Use the formula for pressure (P):
Writing Helper F 8080 lb f
P = = ≈ 6594.1 lb f /f t2 .
A 1.227 f t2
This is equivalent to 6594.1 psia.
For the work done by the gas:
06. Determine the change in volume (ΔV) when the piston is raised by 1.7 feet. The 🚀 Upgrade Sign in
new volume (V2) can be calculated as:
π
V2= (1.252 )(1.7) + V 1 .
4
Assuming V1 is negligible for this calculation, we find:
V 2 ≈ 3.0875 f t3 .
7 Calculate the work done (W) using the formula:
V2
W = P 1 ⋅ ΔV ⋅ ln ( ).
V1
Substituting the values gives:
3.0875
W ≈ 6594.1 lb f /f t2 ⋅ 0.30875 f t3 ⋅ ln ( ) ≈ 1957.95 f t lb f .
(1.227/4π)
Finally, to find the change in potential energy (∆PE):
Homework
08. Use the formula:
∆PE = m ⋅ g ⋅ h = 250 lb m ⋅ 32.169 f t/s2 ⋅ 1.7 f t ≈ 14147.19 f t lb f . Related
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