Simple Harmonic Motion

What is Simple Harmonic Motion?

Simple Harmonic Motion or SHM is defined as a motion in which the restoring force is directly proportional to the displacement of the body from its mean position. The direction of this restoring force is always towards the mean position. The velocity of a particle executing simple harmonic motion is given by,

a (t) = -ω2 x(t)

Here, ω is the angular velocity of the object.

Simple harmonic motion can be referred to as an oscillatory motion in which the velocity of the particle at any position is directly proportional to the displacement from the mean position. It is a special case of oscillatory motion.

All the Simple Harmonic Motions are oscillatory and also periodic but not all oscillatory motions are SHM. Oscillatory motion is likewise called the harmonic motion of all the oscillatory motions wherein the most important one is simple harmonic motion (SHM).

Mass-Spring System


Among the simplest kinds of oscillatory motion is that of a horizontal mass-spring system. If the spring is stretched or compressed through a little displacement x from its mean position, it applies a force F on the mass. According to Hooke’s law, this force is directly proportional to the change in length x of the spring i.e.,

F = – k x

where x is the displacement of the mass from its mean position O, and k is a constant called the spring constant defined as

k = – F/ x

The value of k is a measure of the stiffness of the spring. Stiff springs have a large value of k and soft springs have a small value of k.

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It implies that the acceleration of a mass attached to a spring is directly proportional to its displacement from the mean position. Hence, the horizontal motion of a mass-spring system is an example of simple harmonic motion.

The negative sign indicates that the force applied by the spring is always directed opposite to the displacement of the mass. Since the spring force constantly acts towards the mean position, it is sometimes called a restoring force.

A restoring force constantly pushes or pulls the item performing oscillatory motion towards the mean position. Initially, the mass m is at rest in position mean O and the resultant force on the mass is zero. Suppose the mass is pulled through a distance x up to extreme position A and after that released.

The restoring force exerted by the spring on the mass will pull it towards the mean position O. Due to the restoring force the mass returns, towards the mean position O. The magnitude of the restoring force reduces with the distance from the mean position and becomes zero at O.

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Nevertheless, the mass gains speed as it moves towards the mean position and its speed ends up being optimum at O. Due to inertia the mass does not stop at the mean position O but continues its motion and reaches the extreme position B.


As the mass moves from the mean position O to the extreme position B, the restoring force acting on it towards the mean position steadily increases in strength. Thus, the speed of the mass reduces as it moves towards the extreme position B. The mass finally comes briefly to rest at the extreme position B. Ultimately the mass returns to the mean position due to the restoring force.

This procedure is repeated, and the mass continues to oscillate backward and forward about the mean position O. Such motion of a mass attached to a spring on a horizontal frictionless surface area is referred to as Simple Harmonic Motion (SHM). The time duration T of the simple harmonic motion of a mass ‘m’ connected to spring is given by the below formula:

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Essential features of SHM are summarized as:
  1. A body carrying out SHM always vibrates about a fixed position.
  2. Its acceleration is always directed towards the mean position.
  3. 4. The magnitude of velocity is always directly proportional to its displacement from the mean position i.e., acceleration will be zero at the mean position while it will be optimal at the extreme positions.
  4. Its velocity is optimum at the mean position and zeroes at the extreme positions.

Now we discuss various terms that characterize simple harmonic motion.

  • Vibration: One complete round trip of a vibrating body about its mean position is called one vibration.
  • Time Period (T): The time taken by a vibrating body to finish one vibration is called the time period.
  • Frequency (f): The number of vibrations or cycles of a vibrating body in one second is called its frequency. It is reciprocal of time period i.e., f = 1/T.
  • Amplitude (A): The optimum displacement of a vibrating body on either side from its mean position is called its amplitude.