yibops.blogg.se - Simple harmonic motion definition

However, if the mass is displaced from the equilibrium position, a restoring elastic force which obeys Hooke's law is exerted by the spring. If the system is left at rest at the equilibrium position then there is no net force acting on the mass. (Here the velocity and position axes have been reversed from the standard convention in order to align the two diagrams)Ī simple harmonic oscillator is attached to the spring, and the other end of the spring is connected to a rigid support such as a wall. In Newtonian mechanics, for one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, can be obtained by means of Newton's 2nd law and Hooke's law for a mass on a spring.Simple harmonic motion shown both in real space and phase space. The area enclosed depends on the amplitude and the maximum momentum. Note if the real space and phase space plot are not co-linear, the phase space motion becomes elliptical. If energy is lost in the system, then the mass exhibits damped oscillation.

Thus simple harmonic motion is a type of periodic motion. A net restoring force then slows it down until its velocity reaches zero, whereupon it is accelerated back to the equilibrium position again.Īs long as the system has no energy loss, the mass continues to oscillate.

Therefore, the mass continues past the equilibrium position, compressing the spring. However, at x = 0, the mass has momentum because of the acceleration that the restoring force has imparted. At the equilibrium position, the net restoring force vanishes. When the mass moves closer to the equilibrium position, the restoring force decreases. As a result, it accelerates and starts going back to the equilibrium position. Once the mass is displaced from its equilibrium position, it experiences a net restoring force.

When the system is displaced from its equilibrium position, a restoring force that obeys Hooke's law tends to restore the system to equilibrium.

m −1), and x is the displacement from the equilibrium position (m).įor any simple mechanical harmonic oscillator:.

Where F is the restoring elastic force exerted by the spring (in SI units: N), k is the spring constant ( N Simple harmonic motion provides a basis for the characterization of more complicated periodic motion through the techniques of Fourier analysis.į = − k x, Simple harmonic motion can also be used to model molecular vibration as well. Other phenomena can be modeled by simple harmonic motion, including the motion of a simple pendulum, although for it to be an accurate model, the net force on the object at the end of the pendulum must be proportional to the displacement (and even so, it is only a good approximation when the angle of the swing is small see small-angle approximation). The motion is sinusoidal in time and demonstrates a single resonant frequency. Simple harmonic motion can serve as a mathematical model for a variety of motions, but is typified by the oscillation of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's law. It results in an oscillation which, if uninhibited by friction or any other dissipation of energy, continues indefinitely. In mechanics and physics, simple harmonic motion (sometimes abbreviated SHM) is a special type of periodic motion where the restoring force on the moving object is directly proportional to the magnitude of the object's displacement and acts towards the object's equilibrium position.