A Course of Mathematics for Engineers and Scientists. Volume by Brian H. Chirgwin, Charles Plumpton

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

Sample text

Examples. (i) A uniform rod of length 2a and weight M rests with one end B on a smooth wall and the other end A on the ground where it is smoothly pivoted ; it is kept in equilibrium by a force F acting at B in the plane of the wall. The perpendicular from A to the wall meets it at M and the angle BA M is a. The inclination of MB to the vertical is O. If the normal reaction at the wall is R find the direction of F which makes IF I/IR I a minimum. The forces acting on the rod, resolved in the frame M xyz, are ; at A, {0 2a cosa 0), the reaction {X Y Z}; at B, {2a since sin0 0 2a since cos0}, the forces F, R, {F cosA R F sing} ; at G, {a sinoc sin° a cosa a sing cos0}, the FIG.

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.

A body is pivoted about an axis AB and makes n/27r revolutions per second relative to its bearing. The latter is itself made to revolve with spin co about a fixed axis OZ inclined at an angle 0 to AB, the shortest distance OX between OZ and AB being a. Show that the motion of the body at any instant can be represented by a right-handed screw motion, of pitch (anco sing)/(w2 + n2 + 2con cos0), the axis of this screw intersecting OX at a distance from 0 equal to an (n + co cos 0)/(co2 + n2 + 2wn cos 0).