Solved: The figure depicts the sequence of events in each

Chapter 16, Problem 16.617

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The figure depicts the sequence of events in each cylinder of a four-cylinder internal combustion engine. Each piston moves up and down and is connected by a pivoted arm to a rotating crankshaft. Let and be the pressure and volume within a cylinder at time , where gives the time required for a complete cycle. The graph shows how and vary through one cycle of a four-stroke engine. During the intake stroke (from to ) a mixture of air and gasoline at atmospheric pressure is drawn into a cylinder through the intake valve as the piston moves downward. Then the piston rapidly compresses the mix with the valves closed in the compression stroke (from to ) during which the pressure rises and the volume decreases. At the sparkplug ignites the fuel, raising the temperature and pressure at almost constant volume to . Then, with valves closed, the rapid expansion forces the piston downward during the power stroke (from to ). The exhaust valve opens, temperature and pressure drop, and mechanical energy stored in a rotating flywheel pushes the piston upward, forcing the waste products out of the exhaust valve in the exhaust stroke. The exhaust valve closes and the intake valve opens. Were now back at and the cycle starts again. (a) Show that the work done on the piston during one cycle of a four-stroke engine is , where is the curve in the -plane shown in the figure. [Hint: Let be the distance from the piston to the top of the cylinder and note that the force on the piston is , where is the area of the top of the piston. Then , where is given by . An alternative approach is to work directly with Riemann sums.] (b) Use Formula 16.4.5 to show that the work is the difference of the areas enclosed by the two loops of .