Thermodynamics

EXAMPLE 2–7Power Transmission by the Shaft of a CarDetermine the power transmitted through the shaft of a car when the torqueapplied is 200 N · m and the shaft rotates at a rate of 4000 revolutions perminute (rpm).Solution The torque and the rpm for a car engine are given. The powertransmitted is to be determined.Analysis A sketch of the car is given in Fig. 2–29. The shaft power is determineddirectly fromW # sh 2pn # T 12p2 a4000 83.8 kW1or 112 hp21min b1200 N 1 min# m2a60 s ba 1 kJb1000 N # mDiscussion Note that power transmitted by a shaft is proportional to torqueand the rotational speed.Spring WorkIt is common knowledge that when a force is applied on a spring, the lengthof the spring changes (Fig. 2–30). When the length of the spring changes bya differential amount dx under the influence of a force F, the work done isdW spring F dx(2–27)To determine the total spring work, we need to know a functional relationshipbetween F and x. For linear elastic springs, the displacement x is proportionalto the force applied (Fig. 2–31). That is,F kx1kN2(2–28)where k is the spring constant and has the unit kN/m. The displacement x ismeasured from the undisturbed position of the spring (that is, x 0 whenF 0). Substituting Eq. 2–28 into Eq. 2–27 and integrating yieldW spring 1 2k 1x 2 2 x 2 121kJ2(2–29)where x 1 and x 2 are the initial and the final displacements of the spring,respectively, measured from the undisturbed position of the spring.There are many other forms of mechanical work. Next we introduce someof them briefly.rChapter 2 | 67· ·n · = 4000 rpmT = 200 N • mFIGURE 2–29Schematic for Example 2–7.RestpositionxdxW sh = 2πnTFIGURE 2–30Elongation of a spring under theinfluence of a force.Fn·Torque = FrFIGURE 2–28Shaft work is proportional to thetorque applied and the number ofrevolutions of the shaft.FWork Done on Elastic Solid BarsSolids are often modeled as linear springs because under the action of aforce they contract or elongate, as shown in Fig. 2–32, and when the forceis lifted, they return to their original lengths, like a spring. This is true aslong as the force is in the elastic range, that is, not large enough to causepermanent (plastic) deformations. Therefore, the equations given for a linearspring can also be used for elastic solid bars. Alternately, we can determine