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Mathematical Physics

By Derek F. Lawden

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Example text

47) is This equation is obtained under the assumption that the motion is in the upwards direction. If the motion is in the opposite direction, v has a negative sign, but the sign of mkv2 remains positive. This implies that the sense of the resistance force is not automatically reversed when the sense of the motion changes. 57) in the form dx - v dv g + kv2 [cH. 59) v, we obtain (2. 60) x = 2lk log (g + kv2) + constant. 61) v2 When v = 0, x = h and the particle is at the vertex. 58) in the same manner, we obtain for the x = 2lk log (g kv2) + constant.

A, is termed the amplitude of the motion. (wt + cf>) is called the phase FIG. -Graph of Simple Harmonic Motion of the motion at the instant t and, hence, cf> is known as the initial phase of the motion. cf> can always be made zero by choosing t = 0 to be an instant when x = 0 and is increasing. If we choose t = 0 to be an instant when x = 0 and is decreasing, then cf> = " and x = -a sin wt. " 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.

6) shows that it is calculated by vector addition of these forces. / the same direction. Fn are If the forces Fv F 2, FrG. -Polygon of Forces represented by displacement vectors in the manner indicated in Fig. 1 the displacement vector representing their resultant F forms the closing side of a polygon. This diagram is accordingly called the Polygon of Forces. 2. Work and Power Suppose that a particle moves along a straight line from P to P' under the action of certain forces of which F is one (Fig.

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