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Physics > Oscillations > Simple Harmonic Motion

Simple Harmonic Motion

We see different kinds of motion every day. For example, the motion of the hands of a clock, motion of the wheels of a car, etc. Did you ever notice that these types of motion keep repeating themselves? Well, such motions are periodic in nature, and, one such type of periodic motion is simple harmonic motion (S.H.M.). But, what is S.H.M.? Let’s find out.

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Periodic Motion and Oscillations

Periodic motion is the motion that repeats itself in equal intervals of time. Now, we need to know what periodic motion is, so as to understand simple harmonic motion.

We see many examples of periodic motion in our day-to-day life. For example, the motion of the hands of a clock is a periodic motion, the rocking of a cradle, swinging on a swing, leaves of a tree moving to and fro due to wind breeze, and so on.

In these examples, the particle performs the same set of movements again and again in a periodic motion, and oscillation is one such set of movements. Simple Harmonic Motion is a great example of an oscillatory motion.

Browse more Topics under Oscillations

Download the Cheat Sheet of Oscillations below


Simple Harmonic Motion (S.H.M.)

Simple Harmonic Motion Definition – Simple harmonic motion is the motion in which the object moves to and fro along a line.

For instance, Have you seen a pendulum? When we swing it, it moves to and fro along the same line. These movements are oscillations. Oscillations of a pendulum are an example of simple harmonic motion.

Now, consider there is a spring that is fixed at one end. When there is no force applied to it, it is at its equilibrium position. Now,

  • If we pull it outwards, there is a force exerted by the string that is directed towards the equilibrium position
  • And, if we push the spring inwards, there is a force exerted by the string towards the equilibrium position

Harmonic Motion

In each case, we can see that the force exerted by the spring is towards the equilibrium position, this force is called the restoring force. Now, let the force be F and the displacement of the string from the equilibrium position be x.

Therefore, the restoring force will be F= – kx (the negative sign indicates that the force is in the opposite direction). Here, k is the constant called the force constant. Its unit is N/m in S.I. system and dynes/cm in C.G.S. system.

Learn how to calculate Energy in Simple Harmonic Motion here

Linear Simple Harmonic Motion

Definition – The linear periodic motion of a body in which the restoring force is always directed towards the equilibrium position or mean position and its magnitude is directly proportional to the displacement from the equilibrium position. Also, all simple harmonic motions are periodic in nature, but all periodic motions are not simple harmonic motions.

Now, take the previous example of the string. Let its mass be m. The acceleration of the body will be,

a = F/m = – kx/m = – ω2x

Here, k/m = ω2 (ω is the angular frequency of the body)

Learn the difference between Linear and Damped Simple Harmonic Motion here

Concepts of Simple Harmonic Motion (S.H.M)

  • Amplitude: The maximum displacement of a particle from its equilibrium position or mean position is its amplitude, and its direction is always away from the mean or equilibrium position. Its S.I. unit is the meter, and the dimensions are [L1M0 T0].
  • Period: The time taken by a particle to complete one oscillation is its period. Therefore, the period of S.H.M. is the least time after which the motion will repeat itself. Thus, the motion will repeat itself after nT, where, n is an integer.
  • Frequency: Frequency of S.H.M. is the number of oscillations that a particle performs per unit time. The S.I. unit of frequency is hertz or r.p.s(rotations per second), and its dimensions are [L0M0T-1].
  • Phase: Phase of S.H.M. is its state of oscillation, and the magnitude and direction of displacement of particles represent the phase. Epoch(α) is the phase at the beginning of the motion.

Learn how to find Velocity and Acceleration of Simple Harmonic Motion here

Note: The period of simple harmonic motion does not depend on amplitude or energy or the phase constant.

Difference between Periodic and Simple Harmonic Motion

Periodic Motion Simple Harmonic Motion
In the periodic motion, the displacement of the object may or may not be in the direction of the restoring force. In the simple harmonic motion, the displacement of the object is always in the opposite direction of the restoring force.
Also, the periodic motion may or may not be oscillatory. And, the simple harmonic motion is always oscillatory.
Periodic motion examples are the motion of the hands of a clock, the motion of the wheels of a car, etc. Simple harmonic motion examples: the motion of a pendulum, motion of a spring, etc.


Learn the difference between Periodic and Oscillatory Motion here.

Solved Questions for You

Q: Assertion(A): In simple harmonic motion, the motion is to and fro and periodic
Reason(R): Velocity of the particle V = ω√A2 – x2 where x is displacement as measured from the extreme position
Chose the right answer:

  1. Both a and B are true and R is the correct explanation of A.
  2. Both A and B are true and R is not the correct explanation of A.
  3. A is true, and R is false.
  4. A is false, and R is true.

Solution: c) A is true and R is false. In V = ω√A2 – x2, x measured from the mean position, not from the extreme position, and SHM involves to and fro periodic motion.

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Tamsin AhmedSubhashMridul Bhaskaranju Recent comment authors
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what is the expression for instantaneos amplitude of damped harmonic oscillator

Mridul Bhaskar
Mridul Bhaskar


Patrick raj
Patrick raj



How to find angular velocity and damping constant


you should use antial condition

Tamsin Ahmed
Tamsin Ahmed

Under what condition a Damped Harmonic Motion can be converted to a Simple
Harmonic Motion? Use equations if necessary

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