By D. J. Bell and F. H. George (Auth.)
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It lies in the plane of the paper. The angular velocity of P about O is defined as the rate at which the angle AOP increases. Copy the above diagram onto a sheet of paper because you will need to refer to it again later. What is the angular velocity of a point which moves along the straight line ΟΑΊ 70 Zero. 82 Non-zero.
The angular velocity of P about O is independent of the initial line OA. For example, suppose OB is any other fixed straight line as shown in the above diagram and that OB is taken as initial line. The angular velocity of P about O is the rate at which the angle BOP increases. Since BOP = BOA + AOP, and the angle BOA is constant, it follows that the rate at which the angle BOP increases is the same as the rate at which angle AOP increases. The angular velocity of P about O is thus the same whether OA or OB are the initial lines.
Turn back to 58 66 from 55 For the sake of simplicity the graph is assumed to be solely above the x-axis although this is not necessary. O a M N b The area of interest ABDC is such that OA is of length α,ΟΒ of length b and AC, BD are the ordinates of the points C and D on the curve. Between A and B£n — 1) points are inserted, the distances between successive points not necessarily equal. M and N are two such points in the above diagram. Ordinates such as MP and NQ are drawn from each of these points to the curve, thus dividing the area ABDC into n strips, one of which is MNQP, as shown.