By Brian H. Chirgwin, Charles Plumpton

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Additional resources for A Course of Mathematics for Engineers and Scientists. Volume 6: Advanced Theoretical Mechanics

Example text

The force F is shown in Fig. 16 as two resolutes F cosA = F1and F sin A = F3). Since we do not need to know X, Y, Z we use only the equilibrium condition r (A) = 0. r(A)= AB x (F R) AB x W = 0, {2a since sin° —2a cosa 2a sina cos 0} x {F cos A R F sin A} + + {a sin a sin0 —a cosa a sin a cosO} x (0 0 — W} = 0. {a sing sin0 — a cosa a sin g cos 0} x {2F cosA 2R 2F sin A — W} = 0. In terms of resolutes this leads to the three equations, (2F sinA — W) a cosoc + 2Ra sinoc cos° = 0, 2F a cosA since cos° — (2F sinA — W) a since sin0 = 0, 2Ra sing sin° + 2Fa cosA cosa = 0.

The line is called the central axis (sometimes the instantaneous axis) of the motion. The value of p is not dependent upon the choice of "base point", for, if v is the velocity of any other point of the body, v. co + [co x (r — r,)] . = vA . 46) The scalar product v . w is an invariant for the change of base point. This result concerning the equivalent screw motion is the threedimensional counterpart of the theorem of the existence of the instantaneous centre in plane motion. As the body moves the change of position of the central axis generates one surface in space and another surface in the body; the motion of the body is given by the combination of rolling of the "body-surface" on the space-surface with sliding along the direction of the central axis.

The first equation now gives dt2 d r1 d d iv dv \ d2 v = v ds _ 1 as) A2v d ds d d2 d s2 e% 1 (Az e 2• -c 2/31 = v3 • ds ds e 2 3 2v2 A2e2 — 2 —h. This is the required result. Exercises 1: 1. Show that there are three points on the curve r = au3 + bu2 + + d the osculating planes at which pass through the origin, and that they lie in the plane (r x b). c = 3(r x a). d. 2. A point P moves along a curve in space, the arc from some fixed point P, of the curve up to the point P being s. Prove that, if r is the position vector relative to an origin 0, then r = ,§T; and, if dT/ds = xN, where x z 0, find an expression for the acceleration of P, explaining the significance of the unit vectors T, N and the scalar x.

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