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Learning objectives

By the end of this section, you will be able to:

  • Describe a simple harmonic oscillator.
  • Relate physical characteristics of a vibrating system to aspects of simple harmonic motion and any resulting waves.

The information presented in this section supports the following AP® learning objectives and science practices:

  • 3.B.3.1 The student is able to predict which properties determine the motion of a simple harmonic oscillator and what the dependence of the motion is on those properties. (S.P. 6.4, 7.2)
  • 3.B.3.4 The student is able to construct a qualitative and/or a quantitative explanation of oscillatory behavior given evidence of a restoring force. (S.P. 2.2, 6.2)
  • 6.A.3.1 The student is able to use graphical representation of a periodic mechanical wave to determine the amplitude of the wave. (S.P. 1.4)
  • 6.B.1.1 The student is able to use a graphical representation of a periodic mechanical wave (position versus time) to determine the period and frequency of the wave and describe how a change in the frequency would modify features of the representation. (S.P. 1.4, 2.2)

The oscillations of a system in which the net force can be described by Hooke’s law are of special importance, because they are very common. They are also the simplest oscillatory systems. Simple Harmonic Motion (SHM) is the name given to oscillatory motion for a system where the net force can be described by Hooke’s law, and such a system is called a simple harmonic oscillator    . If the net force can be described by Hooke’s law and there is no damping (by friction or other non-conservative forces), then a simple harmonic oscillator will oscillate 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     X size 12{X} {} . 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; whereas for sound oscillations, they have units of pressure (and other types of oscillations have yet other units). Because amplitude is the maximum displacement, it is related to the energy in the oscillation.

Take-home experiment: shm and the marble

Find a bowl or basin that is shaped like a hemisphere on the inside. Place a marble inside the bowl and tilt the bowl periodically so the marble rolls from the bottom of the bowl to equally high points on the sides of the bowl. Get a feel for the force required to maintain this periodic motion. What is the restoring force and what role does the force you apply play in the simple harmonic motion (SHM) of the marble?

The figure a shows a spring on a frictionless surface attached to a bar or wall from the left side. On the right side of the spring, an object attached to it with mass m, its amplitude is given by X, and X is equal to zero at the equilibrium level. Force F is applied to it from the right side, shown with left direction pointed red arrow and velocity v is equal to zero. A direction point showing the north and west direction is also given alongside this figure as well as with other four figures. In figure b, after the force has been applied the object moves to the left compressing the spring a bit. And the displaced area of the object from its initial point is shown in sketched dots. The F here is equal to zero and the v is max in negative direction. In figure c, the spring has been compressed to the maximum level, and the amplitude is negative X. Now the direction of force changes to the rightward direction, shown with right direction pointed red arrow and the velocity v is zero. In figure d the spring is shown released from the compressed level and the object has moved toward the right side up to the equilibrium level. The F is zero, and the velocity v is maximum. In figure e the spring has been stretched loose to the maximum level and the object has moved to the far right. Now again the velocity here is equal to zero and the direction of force again is to the left hand side, shown here as F is equal to zero.
An object attached to a spring sliding on a frictionless surface is an uncomplicated simple harmonic oscillator. When displaced from equilibrium, the object performs simple harmonic motion that has an amplitude X size 12{X} {} and a period T size 12{T} {} . The object’s maximum speed occurs as it passes through equilibrium. The stiffer the spring is, the smaller the period T size 12{T} {} . The greater the mass of the object is, the greater the period T size 12{T} {} .

Practice Key Terms 3

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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