. Applied calculus; principles and applications . d^UV + 2dudv + ud^v; (2) .-. d^ iuv) =d^UV + 3d^udv + ddud^v + ud^v. (3) The coefficients and exponents of differentiation are according to theBinomial theorem, however far the differentiation is continued; .-. d iuv) = d^w. y + nd^-i udv-]- ^ ^^ ~ ^^ ^^^-2^ d^y + • • • + wdw d*^ y + w d y. 88 DIFFERENTIAL CALCULUS 11. If a;2 + ?/2 = a^, ^3-If^. + |-. = l^ d2ydx2 12. If ,^ = 2 p., g= ^, 14. a2 62 ^dx2 1 !f_»_ __r_ 364_ 70. Circular Motion. — When a point describes a circleof radius r with constant speed v^ it has a constant accelera-tion v^jr d
. Applied calculus; principles and applications . d^UV + 2dudv + ud^v; (2) .-. d^ iuv) =d^UV + 3d^udv + ddud^v + ud^v. (3) The coefficients and exponents of differentiation are according to theBinomial theorem, however far the differentiation is continued; .-. d iuv) = d^w. y + nd^-i udv-]- ^ ^^ ~ ^^ ^^^-2^ d^y + • • • + wdw d*^ y + w d y. 88 DIFFERENTIAL CALCULUS 11. If a;2 + ?/2 = a^, ^3-If^. + |-. = l^ d2ydx2 12. If ,^ = 2 p., g= ^, 14. a2 62 ^dx2 1 !f_»_ __r_ 364_ 70. Circular Motion. — When a point describes a circleof radius r with constant speed v^ it has a constant accelera-tion v^jr directed towards the center of the circle. Let FT be the velocity at P,and PiTi that at Pi. A velocitybeing a directed quantity may berepresented by a vector; that is,by a straight line whose lengthdenotes magnitude and whosedirection is the given from a common origin o,the vectors op and opi are drawnequal to the vectors FT andPiTi, respectively. Since thespeed is constant each vector isincrement, denoted by The Ay. v^ and ppi is the vectoraverage acceleration for the interval of time A^ A^ directed along ppi, and is laid off as pm. As Ai approaches zero. Pi approaches P, and pi approachesp along the circular arc indicated by the dotted line; pmapproaches a vector p^ directed along the tangent to the arc ppi at p. This vector, the lim -r-r L represents the accelera- tion -TT of the point P moving in the circle of radius r; and since the direction is at right angles with the tangent at P,the acceleration is directed towards the center 0, is normalacceleration, therefore, denoted by «„. To find the magni-tude of the normal acceleration a^. since the sectors popiand POPi are similar, the angles at o and 0 being equal, CIRCULAR MOTION 89 arc ppi _ arc PPi arc ppi _ As. op OP V r arcppi V As , and limr^^V^limr^li At r At replacing the arc ppi by its chord, (Art. 22.) ,. rchordppil _ dv _vdsA^<2i L ^i j~ dt ~ r dt dv v^ ,^. •■ « = d« = r (1)
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