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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.
Due to frictional force, the velocity decreases in proportion to the acting frictional force. While in a simple undriven harmonic oscillator the only force acting on the mass is the restoring force, in a damped harmonic oscillator there is in addition a frictional force which is always in a direction to oppose the motion.
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.
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.
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.
The equation is given by ¨ + ˙ + + = (), where the (unknown) function = is the displacement at time t, ˙ is the first derivative of with respect to time, i.e. velocity, and ¨ is the second time-derivative of , i.e. acceleration.
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.
This net force is a restoring force, and the motion of the string can include transverse waves that solve the equation central to Sturm–Liouville theory: [() ()] + () = () where () is the force constant per unit length [units force per area], () is the ...., () is the ...., and are the eigenvalues for resonances of transverse displacement ...