Analytical mechanics for students of physics and engineering . rk done in compressing the fluid from a volume i\to a volume r-2 is W=-J pdv. (1) When the law connecting p and v is given the work donein compressing or expanding a fluid can be found by carry-ing out the integrationindicated in equation (1).During expansion, however,the displacement has thesame direction as the forcewhich causes the expansion;therefore the sign before theintegral is positive. 141. Representation of theWork Done in the PV-Dia-gram.— When the volumeof the expanding fluid isplotted as abscissa and thepressure as ord
Analytical mechanics for students of physics and engineering . rk done in compressing the fluid from a volume i\to a volume r-2 is W=-J pdv. (1) When the law connecting p and v is given the work donein compressing or expanding a fluid can be found by carry-ing out the integrationindicated in equation (1).During expansion, however,the displacement has thesame direction as the forcewhich causes the expansion;therefore the sign before theintegral is positive. 141. Representation of theWork Done in the PV-Dia-gram.— When the volumeof the expanding fluid isplotted as abscissa and thepressure as ordinate, a curve is obtained, which repre-sents, graphically, the law connecting j> and v. Such arepresentation is called a PV-diagram. It i- evidenl fromequation (1) that the area bounded by the curve, the the two vertical lines whose equations are v = >\ andv = Vz, represents the work done in compressing the fluidfrom Vi to Vi. 142. Isothermal Compression of a Gas.— If a gas is com-pressed without changing its temperature the compression. Fig. 96. 176 ANALYTICAL -MECHANICS is called isothermal, in which case the relation between pand v is given by Boyles law, , pv-Jb. (2) Substituting in equation (1) the value of p given by equation(2) we obtain w=-kT- = Hog^- (3) 143. Adiabatic Compression of a Gas.—If no exchange ofheat is allowed between the gas and other bodies while theformer is being compressed the compression is called law which connects p and v in an adiabatic compressionor expansion of a gas may be expressed by the relation pvy = k, (4) where y and k are constants for a given gas. Substitutingin ((iiiation (1) the value of p, which is given by equation(4), we obtain W=-k f* 7- 17- 1 144. Modulus of Elasticity of a Gas. — Let — dv denote thechange in volume due to an increase in the pressure of a gas by an amount dp. Then the stress is dp and the strain - —. v Therefore by Hookes law , . — dvdp = \ V or \=-vp. (6) dv WORK
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