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.

It results in an oscillation that is described by a sinusoid which continues indefinitely (if uninhibited by friction or any other dissipation of energy).

[1] 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.

The motion is sinusoidal in time and demonstrates a single resonant frequency.

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).

Simple harmonic motion can also be used to model molecular vibration.

The motion of a particle moving along a straight line with an acceleration whose direction is always toward a fixed point on the line and whose magnitude is proportional to the displacement from the fixed point is called simple harmonic motion.

[2] In the diagram, a simple harmonic oscillator, consisting of a weight attached to one end of a spring, is shown.

The other end of the spring is connected to a rigid support such as a wall.

If the system is left at rest at the equilibrium position then there is no net force acting on the mass.

However, if the mass is displaced from the equilibrium position, the spring exerts a restoring elastic force that obeys Hooke's law.

For any simple mechanical harmonic oscillator: Once the mass is displaced from its equilibrium position, it experiences a net restoring force.

When the mass moves closer to the equilibrium position, the restoring force decreases.

Therefore, the mass continues past the equilibrium position, compressing the spring.

A net restoring force then slows it down until its velocity reaches zero, whereupon it is accelerated back to the equilibrium position again.

As long as the system has no energy loss, the mass continues to oscillate.

If energy is lost in the system, then the mass exhibits damped oscillation.

The area enclosed depends on the amplitude and the maximum momentum.

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 second law and Hooke's law for a mass on a spring.

Solving the differential equation above produces a solution that is a sinusoidal function:

is the initial speed of the particle divided by the angular frequency,

[A] Each of these constants carries a physical meaning of the motion: A is the amplitude (maximum displacement from the equilibrium position), ω = 2πf is the angular frequency, and φ is the initial phase.

[B] Using the techniques of calculus, the velocity and acceleration as a function of time can be found:

By definition, if a mass m is under SHM its acceleration is directly proportional to displacement.

The above equation is also valid in the case when an additional constant force is being applied on the mass, i.e. the additional constant force cannot change the period of oscillation.

If an object moves with angular speed ω around a circle of radius r centered at the origin of the xy-plane, then its motion along each coordinate is simple harmonic motion with amplitude r and angular frequency ω.

where l is the distance from rotation to the object's center of mass undergoing SHM and g is gravitational acceleration.

The period of a mass attached to a pendulum of length l with gravitational acceleration

This shows that the period of oscillation is independent of the amplitude and mass of the pendulum but not of the acceleration due to gravity,

The linear motion can take various forms depending on the shape of the slot, but the basic yoke with a constant rotation speed produces a linear motion that is simple harmonic in form.