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**

15. , no tendency to cause a translation or to cause a rotation about any axis through any point. 3) where A is an arbitrary point. [It can be proved that if F = 0, and 11 (A) = 0, then T(B) = 0 where B is any other point. See eqn. ] When the forces satisfy these conditions the set is, by definition, in equilibrium. If the resolutes {X Y Z} of F and IL (A) M (A) N (A)} of r (A) are taken in the directions of an arbitrary frame of reference, eqns. 4) as the necessary and sufficient conditions for equilibrium.

14], the instantaneous axis of rotation intersects the sphere in a point of the body which is instantaneously stationary. This is analogous to the instantaneous centre of the plane motion of a lamina. Corresponding to the centrode curves of plane motion the movement of the instantaneous axis of rotation traces out one cone in space, the polhode cone, and another cone in the body, the herpolhode cone. It can be proved that the motion of a body with one point fixed is given by the rolling of the herpolhode cone, taking the body with it, on the polhode cone ; the vertices of the two cones, of course, coincide at the fixed point of the body.

To s and / (s) = 1/1 (r' is a function of s, making b of unit length. 33 r")1 db = / (s)(r' x r"') f' (s)(r' x r"), ds r" x r" 0. since d •i ds / (s) (r' x r'") . r' f (8)(r' x r") . 38) since both triple products contain two identical members r'. 39) This equation defines t and x. 39) holds because the plane containing fi and i, the osculating plane, is the plane for which three intersections with the curve coincide at P. , co, = 0. The component v x of co gives the rate at which the tangent changes direction and so, by comparison with a plane curve, x is called the curvature.

### A Course of Mathematics for Engineers and Scientists. Volume 6: Advanced Theoretical Mechanics by Brian H. Chirgwin, Charles Plumpton

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