Analytical mechanics for students of physics and engineering . es at these points. Then the change in the radial velocityin going from Pi to P2 is obtained by subtracting the radialcomponent of Vi from that of v2. Replace Vi and vs bytheir components along and at right angles to r{ and r-:,respectively, and denote these components by vr;, v andvr, v/):; then it will be seen from the figure that (iV, cos e2 - vP2 sin e2) - Ov, cos €i + vPl sin d) is the total change in the radial velocity. Therefore theradial component of the acceleration is f -Y -t [X cos 62 - vp, sin ei — vr, cos ct - vPl sin
Analytical mechanics for students of physics and engineering . es at these points. Then the change in the radial velocityin going from Pi to P2 is obtained by subtracting the radialcomponent of Vi from that of v2. Replace Vi and vs bytheir components along and at right angles to r{ and r-:,respectively, and denote these components by vr;, v andvr, v/):; then it will be seen from the figure that (iV, cos e2 - vP2 sin e2) - Ov, cos €i + vPl sin d) is the total change in the radial velocity. Therefore theradial component of the acceleration is f -Y -t [X cos 62 - vp, sin ei — vr, cos ct - vPl sin f,1 where t is the time taken by the particle to go from Pi to /.?.But as the points Pi and P approach ! as a limit, the follow-ing substitutions become permissible. 96 ANALYTICAL MECHANICS COS €i = COS 62 Sill ei = d, sin e2 = 62. tv, - vry = dvr, vPl = vPl = vp, ei + «2 = these substitutions in the expression for/r, we obtain/,- limit [(?--^-->(«? + «»)] dVr de dt pdt dt2 r\dt),where 0 is the angle r makes with the a> Fig. .58. (XIV) By similar reasoning we obtain the following expressionsfor the transverse acceleration, that is, the component ofthe acceleration along a perpendicular to the radius vector. * See Appendix Avi. MOTION J = limit \Vr*Sm **+ Vpt C°S e2 ~ ^ ~ Vl Sin €l + ^ C0S 6l^l= limit pvfa + ^+fa-^j r dvp ^ dt u. dr dt + S™ 1 r ddt :»-2«), x\ where co is the angular velocity of the radius vector. PROBLEMS. 1. A particle describes the parabola y = 2 px so that its velocityalong the x-axis is constant and equals //. Find the total velocity and theacceleration. 2. Discuss the motions defined by the following equations derivingthe expressions for the path, velocity, acceleration, and the various com-ponents of the last two: (a) x = at, y = b ^1. (d) .r = at, y = i- (b) x = at, y = bt — \gt-. (e) x = at, y = bean tat. (c) x = aekt, y = bekt. (f) x = a cos cot, y = lit. 3. Express in terms of t the velocity and the accelerati
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