Steam to drive an old-fashioned steam locomotive is supplied...

Steam to drive an old-fashioned steam locomotive is supplied at a constant gauge pressure of $1.75 \times 10^{6} \mathrm{N} / \mathrm{m}^{2}$ (about 250 psi) to a piston with a 0.200 -m radius. (a) By calculating $P \Delta V$, find the work done by the steam when the piston moves $0.800 \mathrm{m}$. Note that this is the net work output, since gauge pressure is used. (b) Now find the amount of work by calculating the force exerted times the distance traveled. Is the answer the same as in part (a)?

Key Concepts

Pressure

Pressure is defined as the force applied per unit area. In thermodynamics and fluid mechanics, it is a key variable that determines how a fluid or gas can perform work, especially in processes where pressure is maintained constant, such as in piston-driven engines.

Gauge Pressure

Gauge pressure is the pressure relative to the ambient atmospheric pressure rather than absolute zero. It is the difference between the measured pressure and the atmospheric pressure, and it is commonly used in engineering applications because many devices, like boilers and hydraulic systems, operate based on this relative pressure.

Work in Constant Pressure Processes (P?V Work)

In a process where the pressure is held constant, the work done by the system can be calculated as the product of the pressure and the change in volume (W = P?V). This formulation is fundamental in thermodynamics for evaluating the energy transferred by fluids when they expand or contract at a steady pressure.

Cross-Sectional Area of a Cylinder

The cross-sectional area of a piston in a cylinder is determined by the area of the circle (A = ?r²). This area is essential in converting a linear displacement of the piston into a change in volume (?V = A × ?x), which then directly relates to the work done by the pressure.

Relationship Between Force, Area, and Pressure

In fluid systems, the force exerted by a fluid on a surface is calculated by multiplying the pressure by the area (F = P × A). This relationship links directly to the mechanical work performed when this force moves an object over a distance, thereby connecting fluid dynamics with classical mechanics.

Equivalence of Work Calculations

When a piston moves under constant pressure, calculating work using P?V and using the force times distance method (F × d) will give equivalent results. This equivalence arises because the force is itself a product of pressure and area, and the displacement over which the force acts can be expressed as a volume change when multiplied by the cross-sectional area.

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