By Housner G & Hudson D
1950 - utilized Mechanics Dynamics, via George W. Housner & Donald E. Hudson (Division of Engineering, California Institute of Technology), D. Van Nostrand Co., Inc., Toronto, N.Y. & London
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6. Find the acceleration of P. Solution. The data for this problem are given in such a way \ 'L.. that a plane polar coordinate 6 ~ c - x x , ) FIG. 6 scriptionis convenient system of the motion. 4)e6 In this problem: r=a+bt2 i. = 2bt Y = 2b (b=ct f=c 4=0 So that : a = [2b - c2(a + btz)]e, + 2bct e+ EXAMPLE3. A particle moves along a path composed of two straight lines connected by a circular arc of radius r, as shown in Fig. 7. The speed along the path is given by S = at. Find the maximum acceleration of the particle.
By the absolute displacement r of the point P is meant the displacement measured with respect to the fixed XYZ system. By MOTION O F A MOVING COORDINATE SYSTEM 39 differentiating this absolute displacement we obtain the absolute velocity i. and the absol~teacceleration r. ' r = Xi' i. = Xi' During these differentiations, the unit vectors it,j', k' are treated as constants, since neither their magnitudes nor their directions change with time. If we wish to express the absolute motion in terms of motion measured in the moving xyz system, we have : where the directions of the i, j , k unit vectors are known with respect to the fixed system.
Soldon. We shall fix the moving xyz coordinate system to the KINEMATICS : T H E DESCRIPTION O F MOTION 42 wheel as shown in Fig. 15. The angular velocity of the coordinate system is o = 4 rad/sec k and the angular acceleration cb = 5 rad/ sec2 k. 51 = -48i In some problems of this type the sum of the first three terms (# + o x (w x p) + cb x p) can be computed more directly by noting that this vector sum represents the absolute acceleration of a fixed point on the moving coordinate system which is coincident with the moving point.