![]() This kind of motion is called simple harmonic motion and the system a simple harmonic oscillator.Ī mass on a spring will trace out a sinusoidal pattern as a function of time, as will any object vibrating in simple harmonic motion. Which when substituted into the motion equation gives:Ĭollecting terms gives B=mg/k, which is just the stretch of the spring by the weight, and the expression for the resonant vibrational frequency: The solution to this differential equation is of the form: Using Hooke's law and neglecting damping and the mass of the spring, Newton's second law gives the equation of motion: Resonant frequency expressionsĪ mass on a spring has a single resonant frequency determined by its spring constant k and the mass m. ![]() Default values will be entered for any missing data, but those values may be changed and the calculation repeated. Then the frequency is f = Hz and the angular frequency = rad/s.Īngular Frequency = sqrt ( Spring constant / ( Mass ) ω = sqrt ( k / m )Īny of the parameters in the equation can be calculated by clicking on the active word in the relationship above. The frequency of simple harmonic motion like a mass on a spring is determined by the mass m and the stiffness of the spring expressed in terms of a spring constant k ( see Hooke's Law): Simple Harmonic Motion Simple Harmonic Motion Frequency Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. We recommend using aĪuthors: Paul Peter Urone, Roger Hinrichs Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses the In fact, the mass m m and the force constant k k are the only factors that affect the period and frequency of simple harmonic motion. For example, a heavy person on a diving board bounces up and down more slowly than a light one. The more massive the system is, the longer the period. Period also depends on the mass of the oscillating system. For example, you can adjust a diving board’s stiffness-the stiffer it is, the faster it vibrates, and the shorter its period. A very stiff object has a large force constant k k, which causes the system to have a smaller period. The period is related to how stiff the system is. Two important factors do affect the period of a simple harmonic oscillator. Because the period is constant, a simple harmonic oscillator can be used as a clock. The string of a guitar, for example, will oscillate with the same frequency whether plucked gently or hard. What is so significant about simple harmonic motion? One special thing is that the period T T and frequency f f of a simple harmonic oscillator are independent of amplitude. The greater the mass of the object is, the greater the period T T. The stiffer the spring is, the smaller the period T T. ![]() The object’s maximum speed occurs as it passes through equilibrium. When displaced from equilibrium, the object performs simple harmonic motion that has an amplitude X X and a period T T. Because amplitude is the maximum displacement, it is related to the energy in the oscillation.įigure 16.9 An object attached to a spring sliding on a frictionless surface is an uncomplicated simple harmonic oscillator. 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). The units for amplitude and displacement are the same, but depend on the type of oscillation. ![]() The maximum displacement from equilibrium is called the amplitude X X. 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 Figure 16.9. 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. They are also the simplest oscillatory systems. 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. Explain the link between simple harmonic motion and waves.By the end of this section, you will be able to: ![]()
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