Analytical mechanics for students of physics and engineering . The components vr and vp may be expressed in terms of xand y by differentiating the equations of transformation r2=a;2+ya (3) A (4) and 6 = tan 82 ANALYTICAL MECHANICS with respect to the time. Differentiating (3) we obtaindrV=dt_x dx y dyr dt r dt= i cos 0 + ?/sin 0. (5) Differentiating (4) we get dd v = r — p dt _rxy-yx x2 + y2 = ycosd-x sin 0. (6) These components satisfy the relation v = Vr2+ r-b\ (7) ILLUSTRATIVE particle describes the motion defined by the equations x = a cos kt, (a) and y = a sin kt. (b) Find the e
Analytical mechanics for students of physics and engineering . The components vr and vp may be expressed in terms of xand y by differentiating the equations of transformation r2=a;2+ya (3) A (4) and 6 = tan 82 ANALYTICAL MECHANICS with respect to the time. Differentiating (3) we obtaindrV=dt_x dx y dyr dt r dt= i cos 0 + ?/sin 0. (5) Differentiating (4) we get dd v = r — p dt _rxy-yx x2 + y2 = ycosd-x sin 0. (6) These components satisfy the relation v = Vr2+ r-b\ (7) ILLUSTRATIVE particle describes the motion defined by the equations x = a cos kt, (a) and y = a sin kt. (b) Find the equation of the path, the velocity at any instant, and the com-ponents of the latter. Squaring and adding (a) and (b) we eliminate t and obtainX2 -f if = o* Y for the equation of the (a), we have . dxX = dJ = —ha si i kt = -ky. Differentiating (b), we obtain . dy = ka ens kt= MOTION 83 Therefore v = Vx- + y- = k Vx- + if= ka. Thus the particle describes a circle with a constant speed ka. The direc-tion of the velocity at any instant is given by the relation tan 6 = I The components vr and vp may be obtained at once by remembering, (1)that the radius vector is constant: , f = 0, (2) that it is always normalto the path: , rdd = ds. Therefore dr n v* = Tt = °»at and dd dsTdt=dt=V ka. 82. Velocity of a Particle Relative to Another Particle in Motion.—Consider the motion of a particle Pi, Fig. 52, with Y
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Keywords: ., bookcentury1900, bookdecade1910, bookpublishernewyo, bookyear1913
