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

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

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42), by ox = oxA (R g). The term (5 (114) is a small rotation which can be written as 60 x (r — rA), using eqn. 32). 44a) Sr = SrA + 60 x (r rA). — [This result may also be obtained from eqn. ] Examples. (i) Two wheels, of radii a and b and centres A and B respectively, are freely mounted at the ends of an axle AB with their planes at right angles to the axle. They are placed with their rims resting on a rough horizontal plane on which they roll without slipping, the distance apart of their points of contact being c.

We use the frame of reference A l23shown in Fig. 11 with A S3 along the axle and A Sl in a vertical plane passing through C, where the wheel with centre A touches the horizontal plane. The phrase "spin about an axis" means the resolute of the angular velocity vector along this axis. The three "rigid bodies" here have angular velocities : the frame of reference, angular velocity 0 = {01 02 03} the wheel at A, angular velocity w = {01 02co} ; angular velocity co' = {61 02 col. the wheel at B, [Since the wheels are mounted on the axle their angular velocities can differ from 0 only in the resolute along A It is intuitively apparent that the angular velocity of the frame of reference is directed along the vertical, which is given by the unit vector A = {cos oc 0 — sin oc}.

4) as the necessary and sufficient conditions for equilibrium. ] The reader will recall, from the last chapter, that a rigid body has 6 degrees of freedom and so needs 6 conditions to fix its position. 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.