# Simple Harmonic motion

The simplest kind of oscillation occurs when the restoring force F_{x }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 F_{x} 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 F_{x} = k_{x}. 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.

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 = d^{2 }x/dt^{2} = F_{x}/m of a body in SHM is

The minus sign means that, in SHM, the acceleration and displacement always have opposite signs. This acceleration is not constant, so don’t even think of using the constant-acceleration equations from Chapter 2. We’ll see shortly how to solve this equation to find the displacement x as a function of time. A body that undergoes simple harmonic motion is called a harmonic oscillator.

Why is simple harmonic motion important? Not all periodic motions are simple harmonic; in periodic motion in general, the restoring force depends on displacement in a more complicated way than in Eq. (14.3). But in many systems the restoring force is approximately proportional to displacement if the displacement is sufficiently small (Fig. 14.4). That is, if the amplitude is small enough, the oscillations of such systems are approximately simple harmonic and therefore approximately described by Eq. (14.4). Thus we can use SHM as an approximate model for many different periodic motions, such as the vibration of a tuning fork, the electric current in an alternating-current circuit, and the oscillations of atoms in molecules and solids.

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