By Derek F. Lawden
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Extra resources for A Course in Applied Mathematics, Vol. 1 and 2
4) F being measured in units of force appropriate to those employed for mass and acceleration. 4) reveals that the Earth's attraction on a particle at its surface is proportional to that particle's mass. This result is included in a further law due to Newton, which we shall state in Chapter 4 (p. 102) . If a number of forces act upon a particle the acceleration produced is found to be the vector sum of the accelerations each would cause if acting separately. This fact is an additional law of motion, whose validity can be checked only by experiment.
Depends only upon the material and cross-sectional area of the string. When the particle hangs in equilibrium (Fig. , T = mg. 35) mg = [LC. 37) If the particle is now displaced vertically from its equilibrium position and released, it will oscillate in a vertical line. Let 0 re present the equilibrium position of the particle on this line and x the distance of P vertically below 0 at time t (Fig. 7 (b) ) . 38) 34 A CO URSE IN APPL IED MATHEMAT ICS [CH. 37). , where w2 = fLfm. 40) implies that the particle's motion is simple harmonic of period 2rr:/w = 2rr:vmffL .
Employing the notation of vector algebra, if d represents the dis-+ placement PP', the work done by F can be written as the scalar product F d If F is the resultant of a number of forces F1 , F2, . 6). We have shown, therefore, that the sum of the works done by a number of forces is equal to the work done by their resultant. 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.
A Course in Applied Mathematics, Vol. 1 and 2 by Derek F. Lawden