Simple harmonic motion (SHM)

Simple harmonic motion (SHM)

Simple harmonic motion (SHM) is a fundamental phenomenon in kinematics that describes the oscillation of an object around an equilibrium point. Some illustrative examples of this phenomenon include the movement of a simple pendulum and the oscillation of a particle attached to a previously compressed spring.

This type of movement manifests itself as a periodic back-and-forth oscillation, where a body moves from one side to the other of its equilibrium position in uniform time intervals. 

When the trajectory of the particle is rectilinear, the oscillation in a SHM. It involves alternating movements of moving away and approaching to a central point in its trajectory. In this context, the position of the particle as a function of time exhibits the characteristic shape of a sinusoid. In this type of movement, the force acting on the particle is directly proportional to its displacement with respect to the central point, always directed towards the latter.

Types of movement: oscillations and vibrations

Oscillations and vibrations refer to the repetitive pattern of motion that an object experiences around an equilibrium position. Both terms are commonly used in physics to describe different aspects of simple harmonic motion:


Oscillations refer to the movement of an object back and forth around a central or equilibrium position. In this type of motion, the object oscillates along a path that varies symmetrically on either side of the equilibrium position in a manner that may be periodic or nonperiodic.

For example, in the case of a simple pendulum, the oscillation occurs as the pendulum moves back and forth.


Vibrations are oscillations that occur at a specific frequency. In this context, vibrations refer to the repetitive and periodic movement of the object around its equilibrium position. It is often used to describe high-frequency oscillations that are difficult to perceive individually. Vibration generally involves a rapid, repetitive oscillation of an object or system around an equilibrium point.

Dynamic analysis: the restoring force

At the core of simple harmonic motion is a restoring force proportional to the displacement of the object, but in the opposite direction. This type of movement occurs when an object moves away from its equilibrium position and experiences a force that returns it to that position.

In the case of our previous examples, in a simple pendulum, the restoring force acting on it is related to the gravitational force. The restoring force on a simple pendulum is provided by the gravitational force component. On the other hand, in the case of the spring, this force is exerted by the spring itself.

The most basic form of this force is expressed by Hooke's law:



  • F is the restoring force.
  • k is the spring constant, which represents the stiffness of the system.
  • x is the displacement from the equilibrium position.

Equation of Simple Harmonic Motion (SHM)

The position of an object in harmonic motion as a function of time is described by the following equation:

x ( t )= A ⋅cos( ωt + ϕ )


  • x ( t ) is the position as a function of time.
  • A is the amplitude, the maximum distance from the equilibrium position.
  • ω is the angular frequency, related to the ordinary frequency f by ω =2 πf .
  • ϕ is the initial phase, which determines the position of the object at the beginning of the oscillation.

In the definition of simple harmonic motion, numerous concepts appear that we may not understand perfectly. Below we attach a glossary of terms along with a brief description of their meaning:

  • Amplitude (A): The maximum distance from the equilibrium position to the farthest point of the oscillation.
  • Ordinary Frequency (f): The number of complete oscillations per unit of time.
  • Angular Frequency (ω): The measure of the speed with which oscillations occur, related to the ordinary frequency (f) by the formula ω=2πf.
  • Period (T): The period of a SHM is the time it takes to complete a complete oscillation. It is the inverse of the frequency T = 1/f.
  • Initial Phase (ϕ): A parameter in the SHM equation that determines the initial position of the oscillation relative to time.
  • Hooke's Law: The law that describes the restoring force in the linear sense.
  • Displacement (x): The distance from the equilibrium position to the current position of the object.
  • Spring constant (k): A parameter that determines the stiffness of a spring in Hooke's law.
  • Oscillation: The movement of an object from one side to the other of its equilibrium position in a repetitive pattern.
  • Rectilinear path: A type of harmonic motion in which the particle oscillates along a straight line.
  • Sinusoid: The graphical representation of position versus time in a SHM, which follows a sinusoidal curve.

5 Examples of simple harmonic motion

motorcycle shock absorberSimple harmonic motion (SHM) is present in numerous everyday phenomena that we experience in our daily lives. Here are some examples:

  1. Clock pendulum: The clock pendulum is a classic example of SHM. When the pendulum moves from its equilibrium position, the gravitational force returns it to that position, generating regular oscillations that mark time.
  2. Swing Oscillations: When someone swings on a swing, they experience simple harmonic motion. The gravitational force provides the restoring force, and the swing swings back and forth.
  3. Movement of a spring: If you hang an object on a spring and compress or stretch it, the object will perform a SHM. As the spring stretches or compresses, the restoring force of the spring (described by Hooke's law) will cause the object to oscillate around its equilibrium position.
  4. Musical String Vibration: When a musical string is played, it vibrates in a sinusoidal pattern. The tension in the string provides the restoring force, and the string oscillates up and down producing sound.
  5. Motion of a piston in an automobile engine: The piston in an automobile engine moves back and forth, which can be approximated by simple harmonic motion. Combustion in the engine cylinder provides the force that drives the movement of the piston, which oscillates repeatedly.
Publication Date: March 13, 2024
Last Revision: March 13, 2024