By Brian H. Chirgwin, Charles Plumpton

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**Extra info for A Course of Mathematics for Engineers and Scientists. Volume 6: Advanced Theoretical Mechanics **

**Example text**

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 two extreme cases of the screw motion are: (a) p = 0; v .

56) CA w1 (w1 w2cos2a) When a = 0, n/2 this result is equivalent to eqn. 53) above. Miscellaneous Exercises 1 1. Prove that the angle of rotation a of the resultant rotation defined by Euler's angles, eqn. 17), is given by cos ia = coslO cos i(q) p). 2. A right circular cone of vertical angle 2a rolls without slipping on a rough plane, its vertex being at a fixed point in the plane. 12 is the angular velocity of the cone about its axis, find the angular velocities of (a) the generator of contact, (b) the cone about this generator.

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.