# Simple Harmonic motion

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The simplest kind of oscillation occurs when the restoring force Fx is directly proportional to the displacement from equilibrium x. This happens if the spring in Figs. 14.1 and 14.2 is an ideal one that obeys Hooke’s law (see Section 6.3). The constant of proportionality between Fx and x is the force constant k. On either side of the equilibrium position, Fx and x always have opposite signs. In Section 6.3 we represented the force acting on a stretched ideal spring as Fx = kx. The x-component of force the spring exerts on the body is the negative of this, so,

This equation gives the correct magnitude and sign of the force, whether x is positive, negative, or zero (Fig. 14.3). The force constant k is always positive and has units of N>m (a useful alternative set of units is kg/s 2 ). We are assuming that there is no friction, so Eq. (14.3) gives the net force on the body.

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When the restoring force is directly proportional to the displacement from equilibrium, as given by Eq. (14.3), the oscillation is called simple harmonic motion (SHM). The acceleration ax = d2 x/dt2 = Fx/m of a body in SHM is

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