By Derek F. Lawden
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Extra info for A Course in Applied Mathematics, Vol. 1 and 2
28) with respect to t. 34) = aw. v v = aw cos (wt + if>), Vmax. , when P passes through 0 in the positive direction. , see Example 3 below) . Practically, the simplest way of causing a body to execute SHM is by connecting it to a fixed point by an elastic support. Thus, suppose that a particle P of mass m hangs freely from a fixed point A by an 2] NE WTON ' S LA W S . RECT IL INEAR MOTION 33 elastic string of natural length l (Fig. 7). X mg t T p mg FIG. -Vertical Oscillations on an Elastic String Then, if e is the extension beyond the natural length and T is the tension produced, proportional to its tension.
The motion is called Simple Harmonic Motion (SHM) and the time of one complete oscillation is the period. Denoting the period by T, we have T = 2rtcu . 29) or - 1 - 2] NE WTON 'S LAWS . RE CTIL INEAR MOT ION 31 The number of complete oscillations taking place in unit time is called the frequency f. Thus (>) 1 ! , 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.
Einstein's criticism of this assumption led him to develop the Special Theory of Relativity, and it is now uni versally admitted that the existence of such an absolute frame cannot be demonstrated. A more mature interpretation of the law must therefore be presented. If the centre of the Sun is adopted as the origin of a frame of re ference, the axes of which do not rotate against the background of the extra-galactic nebulce, then we may accept it as an observable fact that those stars of our galaxy which are widely separated from their neigh bours follow straight-line tracks at constant speeds over very long periods of time.
A Course in Applied Mathematics, Vol. 1 and 2 by Derek F. Lawden