By Derek F. Lawden
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Additional info for A Course in Applied Mathematics, Vol. 1 and 2
If F is the resultant of a number of forces, since it remains constant in magnitude and direction over PP', provided this arc is small, the work it does over this element is the sum of the works done by the individual forces. Summing over all the elements as, we conclude that this statement is also valid for the whole arc AB. If P has position vector r and P' has position vector r + ar, the dis placement PP' along the curve A B is denoted by ar and the line integral (2. 7) may be written in vector notation thus : W = LsF · dr.
And x = a cos wt. 31) = A sin wt + B cos wt, where A = a cos cf>, B = a sin cf> and are therefore also of the nature of constants of integration. 24). 28) indicates that SHM may be regarded as the pro jection on a straight line of a uniform circular motion. Thus, consider a point Q which moves around a circle of radius a and centre 0 with 32 A COURSE IN A P PLIED MATHEMAT I C S [CH. uniform angular velocity w in a clockwise direction (Fig. 6). Let A A ', BB' be perpendicular diameters of the circle and let QP be FrG.
Motion Along the x-Axis when P is on one side of 0 and negative when P lies on the other side . If v is the speed of P, v = dxjdt = i and v is positive if x is increasing. Let F be the magnitude of the force acting on P, F being given a positive sign if it acts in the direction in which x increases. We shall show how the motion of P may be determined in the three cases when F is given as a function of (i) t, (ii) x and (iii) v. (1) F is a Function of t The acceleration of P is dvjdt. 12) F = m dv dt' m being a constant, this implies that dvjdt is a known function of t.