By Derek F. Lawden
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Extra resources for A Course in Applied Mathematics, Vol. 1 and 2
Newton's Laws of Motion First Law of Motion may be rendered as follows : A body continues in its state of rest or of uniform motion in a straight line, unless acted upon by forces. Newton's We shall identify the " body " 'Yith the " particle " defined in the previous chapter. The motion of a particle can be assessed only against the background of a frame of reference and it is clear from the terms in which the law is formulated that this frame is supposed to be " at rest ". Newton assumed that it was possible to distinguish from among the infinity of possible frames of reference one which was in a state of absolute rest.
14) yields the value of v appropriate to this instant. 15) We shall normally find it convenient to choose 0 to be the position of the particle at time t = 0. 15} is satisfied by these values. As a particular case, F may be constant. The acceleration is then also constant and equal to f, say. Thus, = = dv = f. 16) v = u + jt, where v = u at t = 0. 17) x = ut + tft2 , having assumed that x = 0 at t 0. , v2 = u2 + 2jx. 18) An electric train of mass M tons starts moving along a straight, level track from rest.
P, � eo/1 cota. _ e-o/1 cota. v. • t = constant - � tan IX e-o/1 cot a. v. _tan et . We can now v. Pcot a in equation (ii) to obtain a tan et . _ v. tan a, v---+ oo. _ v. tan a. A particle, initially at rest at the origin, moves in a plane with an acceleration that is the resultant of a constant radial acceleration f outwards and a variable acceleration equal to 2w times the velocity in a v direction perpendicular to the velocity, where w is constant. Show that its angular velocity about the origin is equal to w f and find the polar equation of its path, taking the direction of f at the origin as the initial line.