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In mechanics and physics, simple harmonic motion (sometimes abbreviated as SHM) is a special type of periodic motion an object experiences by means of a restoring force whose magnitude is directly proportional to the distance of the object from an equilibrium position and acts towards the equilibrium position.
The force responsible for restoring original size and shape is called the restoring force. [1] [2] An example is the action of a spring. An idealized spring exerts a force proportional to the amount of deformation of the spring from its equilibrium length, exerted in a direction oppose the deformation.
When a spring is stretched or compressed by a mass, the spring develops a restoring force. Hooke's law gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length: = (), where F is the force, k is the spring constant, and x is the displacement of the mass with respect to the equilibrium ...
In harmonic oscillators, the restoring force is proportional in magnitude (and opposite in direction) to the displacement of x from its natural position x 0. The resulting differential equation implies that x must oscillate sinusoidally over time, with a period of oscillation that is inherent to the system.
According to this formula, the graph of the applied force F s as a function of the displacement x will be a straight line passing through the origin, whose slope is k. Hooke's law for a spring is also stated under the convention that F s is the restoring force exerted by the spring on whatever is pulling its free end.
Simple harmonic motion – motion in which the body oscillates in such a way that the restoring force acting on it is directly proportional to the body's displacement. Mathematically Force is directly proportional to the negative of displacement. Negative sign signifies the restoring nature of the force. (e.g., that of a pendulum).
The systems where the restoring force on a body is directly proportional to its displacement, such as the dynamics of the spring-mass system, are described mathematically by the simple harmonic oscillator and the regular periodic motion is known as simple harmonic motion.
The animations below depict the motion of a simple (frictionless) pendulum with increasing amounts of initial displacement of the bob, or equivalently increasing initial velocity. The small graph above each pendulum is the corresponding phase plane diagram; the horizontal axis is displacement and the vertical axis is velocity. With a large ...