where x is the instantaneous velocity of the mass.
The force in the spring is kx; work done by the spring is kx Sx. The potential energy is the work done by this force and is given by
The total energy in the system is a constant. Thus
Differentiating with respect to x, we get mxx + kxx = 0 (2.59)
Since x cannot be zero for all values of t, we get mx + kx = 0 (2.60)
which has the same form as Eq. (2.52). This differential equation has a solution of the form x = A sin(wt + a) (2.61)
where A is the maximum displacement and wA is the maximum velocity. Maximum kinetic energy is given by
Maximum potential energy is
or which is the natural frequency of the simple oscillator. This method, in which the natural frequency is obtained by equating maximum kinetic energy and maximum potential energy, is known as Rayleigh's method.
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