SIMPLE HARMONIC MOTION
The very simplest kind of oscillation occurs when the restoring force F is directly proportional to the displacement from equilibrium x. This happens if the spring in Fig. 13-1 is an ideal one that obeys Hooke’s law. The constant of proportionality between F and x is the force constant k. (You may want to review Hooke’s law and the definition of the force constant in Section 6-4.) On either side of the equilibrium position, F and x always have opposite signs. In Section 6-4 we represented the force acting on a stretched ideal spring as F = kx. The force the spring exerts on the body is the negative of this, so the x-component of force F on the body is This equation gives the correct magnitude and sign of the force, whether x is positive, negative, or zero. The force constant k is always positive and has units of N/m (a useful alternative set of units is kgll). We are assuming that there is no friction, so Eq. (13-3) gives the net force on the body.
When the restoring force is directly proportional to the displacement from equilibrium, as given by Eq. (13-3), the oscillation is called simple harmonic motion, abbreviated. The acceleration a = d2x1dt2 = Flm of a body in is given by 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.
Why is simple harmonic motion important? Keep in mind that 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. (13-3). But in many systems the restoring force is approximately proportional to displacement if the displacement is sufficiently small (Fig. 13-2). That is, if the amplitude is small enough, the oscillations of such systems will be approximately simple harmonic and therefore approximately described by Eq. (13-4). Thus we can use SHM as an approximate model for many different periodic motions, such as the vibration of the quartz crystal in a watch, the motion of a tuning fork, the electric current in an alternating-current circuit, and the vibrations of atoms in molecules and solids,