By Derek F. Lawden

ISBN-10: 0340045868

ISBN-13: 9780340045862

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Additional info for A Course in Applied Mathematics, Vol. 1 and 2

Example text

7) may be written in vector notation thus : W = LsF · dr. Suppose that the curve AB and the force F lie in the plane of rect­ angular axes Ox, Oy and let (X, Y) be the components of F parallel to the axes. If P is the point (x, y) and P' is the point (x + ax, y + ay), ar has components (ax, ay). Hence F · ar = Xax + Yay and W = LB (Xdx + Ydy) . 8) A particle can move in the plane of rectangular axes Ox, Oy. When its coordinates are (x, y), the components of a certain force applied to it are (ky2, kx2) .

Now consider the general case when a particle moves along any curve from A to B under the action of forces of which F is one (Fig. 3) . · . · · · · F B s A FIG. -Work Done by a Variable Force F will be supposed to vary both in magnitude and direction. Let P be any point on the curve and let the arc AP be denoted by s. Let the arc P P' be as. Let e be the angle between the direction of F when -+ -+ acting at P and the tangent PT to the curve at P. PT is taken in the same sense as the motion. As the particle moves from P to P', provided as is sufficiently small, we may assume that F does not vary appreciably in magnitude or direction.

Suppose that F remains constant in magnitude and direction during the motion , F L,, '�,, p p (b) (a) FIG. -Work Performed by Force e being the angle made by the direction of the force with that of the displacement PP'. e is taken to be acute in case (a) and obtuse in case (b). The work done by this force during the displacement is defined to be the quantity F . P P' cos e, and is therefore negative if e is obtuse and zero if e is a right angle. F cos e is the resolute of F in the direction of the displacement PP', and hence the work done is equal to the resolute of the force in the direction of the displacement multiplied by the displacement.

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A Course in Applied Mathematics, Vol. 1 and 2 by Derek F. Lawden

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