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Characteristics of simple harmonic motion

A very common type of periodic motion is called simple harmonic motion (SHM)    . A system that oscillates with SHM is called a simple harmonic oscillator    .

Simple harmonic motion

In simple harmonic motion, the acceleration of the system, and therefore the net force, is proportional to the displacement and acts in the opposite direction of the displacement.

A good example of SHM is an object with mass m attached to a spring on a frictionless surface, as shown in [link] . The object oscillates around the equilibrium position, and the net force on the object is equal to the force provided by the spring. This force obeys Hooke’s law     F s = k x , as discussed in a previous chapter.

If the net force can be described by Hooke’s law and there is no damping (slowing down due to friction or other nonconservative forces), then a simple harmonic oscillator oscillates with equal displacement on either side of the equilibrium position, as shown for an object on a spring in [link] . The maximum displacement from equilibrium is called the amplitude ( A )    . The units for amplitude and displacement are the same but depend on the type of oscillation. For the object on the spring, the units of amplitude and displacement are meters.

The motion and free body diagrams of a mass attached to a horizontal spring, spring constant k, at various points in its motion. In figure (a) the mass is displaced to a position x = A to the right of x =0 and released from rest (v=0.) The spring is stretched. The force on the mass is to the left. The free body diagram has weight w down, the normal force N up and equal to the weight, and the force F to the left. (b) The mass is at x = 0 and moving in the negative x-direction with velocity – v sub max. The spring is relaxed. The Force on the mass is zero. The free body diagram has weight w down, the normal force N up and equal to the weight. (c) The mass is at minus A, to the left of x = 0 and is at rest (v =0.) The spring is compressed. The force F is to the right. The free body diagram has weight w down, the normal force N up and equal to the weight, and the force F to the right. (d) The mass is at x = 0 and moving in the positive x-direction with velocity plus v sub max. The spring is relaxed. The Force on the mass is zero. The free body diagram has weight w down, the normal force N up and equal to the weight. (e) the mass is again at x = A to the right of x =0 and at rest (v=0.) The spring is stretched. The force on the mass is to the left. The free body diagram has weight w down, the normal force N up and equal to the weight, and the force F to the left.
An object attached to a spring sliding on a frictionless surface is an uncomplicated simple harmonic oscillator. In the above set of figures, a mass is attached to a spring and placed on a frictionless table. The other end of the spring is attached to the wall. The position of the mass, when the spring is neither stretched nor compressed, is marked as x = 0 and is the equilibrium position. (a) The mass is displaced to a position x = A and released from rest. (b) The mass accelerates as it moves in the negative x -direction, reaching a maximum negative velocity at x = 0 . (c) The mass continues to move in the negative x -direction, slowing until it comes to a stop at x = A . (d) The mass now begins to accelerate in the positive x -direction, reaching a positive maximum velocity at x = 0 . (e) The mass then continues to move in the positive direction until it stops at x = A . The mass continues in SHM that has an amplitude A and a period T . The object’s maximum speed occurs as it passes through equilibrium. The stiffer the spring is, the smaller the period T . The greater the mass of the object is, the greater the period T .

What is so significant about SHM? For one thing, the period T and frequency f of a simple harmonic oscillator are independent of amplitude. The string of a guitar, for example, oscillates with the same frequency whether plucked gently or hard.

Two important factors do affect the period of a simple harmonic oscillator. The period is related to how stiff the system is. A very stiff object has a large force constant ( k )    , which causes the system to have a smaller period. For example, you can adjust a diving board’s stiffness—the stiffer it is, the faster it vibrates, and the shorter its period. Period also depends on the mass of the oscillating system. The more massive the system is, the longer the period. For example, a heavy person on a diving board bounces up and down more slowly than a light one. In fact, the mass m and the force constant k are the only factors that affect the period and frequency of SHM. To derive an equation for the period and the frequency, we must first define and analyze the equations of motion. Note that the force constant is sometimes referred to as the spring constant .

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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