Simple Harmonic Motion is a fundament concept in the study of motion, especially oscillatory motion; which helps us understand many physical phenomena around like how strings produce pleasing sounds in a musical instrument such as the sitar, guitar, violin, etc., and also, how vibrations in the membrane in drums and diaphragms in telephone and speaker system creates the precise sound. Understanding Simple Harmonic Motion is key to understanding these phenomena.

In this article, we will grasp the concept of Simple Harmonic Motion (SHM), its examples in real life, the equation, and how it is different from periodic motion.

Table of Content

**Simple Harmonic Motion Definition (SHM Definition)**

**Simple Harmonic Motion Definition (SHM Definition)**

Simple harmonic motion is an oscillatory motion in which the acceleration of particle at any position is directly proportional to its displacement from the mean position.

As SHM is an example of Oscillatory Motion. Simple Harmonic Motions (SHM) are all oscillatory and periodic, but not all Oscillatory or Periodic motions are SHM. Oscillatory motion is also referred to as Harmonic Motion and out of all Harmonic Motions, the most important one to study is Simple Harmonic Motion (SHM). Some characteristics of SHM are as follows:

- SHM is a type of Periodic and Oscillatory Motion.
- There is always a restoring force acting on an object in SHM, which always acts in the opposite direction to the displacement of the object from the mean position.
- The amplitude in the SHM remains constant throughout the motion of the object.
- The acceleration of the object is directly proportional to the displacement of the object from its mean position.
- The velocity of the object is maximum at the equilibrium position.
- The total energy in SHM remains conserved, as there is always a conversion of kinetic and potential energy happening throughout the motion.

### Examples of Simple Harmonic Motion(SHM)

There are a lot of examples of Simple Harmonic Motion around us, we just need to see them from the perspective of SHM. From swings in the park to the motion of the cantilever, all are examples of SHM. The following illustration shows different examples of Simple Harmonic Motion.

**Terminology Related to Simple Harmonic Motion**

**Terminology Related to Simple Harmonic Motion**

There are various terminologies related to SHM (Simple Harmonic Motion) some of which are explained as follows:

### Mean Position

In Simple Harmonic Motion, the position of the object where there is no restoring force acting on it is the mean position. In other words, the point about which the object moves between its extreme position is called the mean position of the object. The mean position is sometimes referred to as Equilibrium Position as well.

**Amplitude**

**Amplitude**

The amplitude of a particle in SHM is its maximum displacement from its equilibrium or mean position, and as displacement is a vector quantity, its direction is always away from the mean or equilibrium position. The SI unit of amplitude is the meter and all the other units of length can also be used for this.

**Frequency**

**Frequency**

The frequency of SHM is the number of oscillations performed by a particle per unit of time. SI unit of frequency is Hertz or r.p.s. (rotations per second), and is given by:

- Frequency

f = 1/ T

- Angular Frequency

ω = 2πf = 2π/T

**Time Period**

**Time Period**

For a particle performing SHM, the time period is the amount of time it takes to complete one complete oscillation. As a result, the time period or simply period of SHM is the shortest time before the motion repeats itself.

T = 2π/ωwhere ω is the Angular frequency and T is the Time period.

**Phase**

**Phase**

The phase of SHM represents the magnitude and direction of particle displacement at any instant of the motion which is its state of oscillation.

The expression for a particle’s position as a function of time and angular frequency is as follows:

x = A sin (ωt + ϕ)where

is the phase of particle.(ωt + ϕ)

**Phase Difference**

**Phase Difference**

For two particles performing SHM, the phase difference is defined as the difference between the total phase angles of those particles. Phase Difference is denoted by Δϕ. Mathematically the phase difference is defined as the difference between the total phase angles of two particles moving in simple harmonic motion with respect to the mean position.

For example, for two particles performing SHM with the same angular frequency with displacement functions, x_{1} = A sin (ωt + ϕ_{1}) and x_{2} = A sin (ωt + ϕ_{2}). The phase difference is given by

Δϕ = ϕ_{1}– ϕ_{2}

When two vibrating particles with the same angular frequency, are in the same phases if the phase difference between them is an even multiple of π i.e.,

Δϕ = nπWhere, n = 0, 1, 2, 3, 4, . . .

Two vibrating particles with the same angular frequency, are said to be in opposite phases if the phase difference between them is an odd multiple of π i.e.,

Δϕ = (2n + 1)πWhere, n = 0, 1, 2, 3, 4, . . .

## Types of Simple Harmonic Motion (SHM)

There are two types of SHM, which is:

**Linear Simple Harmonic Motion****Angular Simple Harmonic Motion**

**Linear Simple Harmonic Motion**

**Linear Simple Harmonic Motion**

When a particle moves back and forth along a straight line around a fixed point (called the equilibrium position), this is referred to as ** Linear Simple Harmonic Motion.** Some examples of Linear SHM include the oscillation of a liquid column U-tube, the motion of a simple pendulum with very small displacements, and the vertical small vibration of a mass carried by elastic string.

**Conditions for Linear Simple Harmonic Motion**

**Conditions for Linear Simple Harmonic Motion**

The restoring force or acceleration acting on the particle must always be proportional to the particle’s displacement and directed toward the equilibrium position.

F ∝ – X

a ∝ -xwhere

is the Restoring ForceFis the Displacement of Particle from Equilibrium PositionXis the Accelerationa

**Angular Simple Harmonic Motion**

**Angular Simple Harmonic Motion**

An angular simple harmonic motion occurs when a system oscillates angularly with respect to a fixed axis. The displacement of the particle in angular simple harmonic motion is measured in terms of angular displacement. The torsional pendulum is one example of Angular SHM.

**Conditions for Angular Simple Harmonic Motion**

**Conditions for Angular Simple Harmonic Motion**

The restoring torque (or) angular acceleration acting on the particle should always be proportional to the particle’s angular displacement and oriented towards the equilibrium position.

T ∝ -θ

α ∝ -θwhere

is TorqueTis the Angular Displacementθis the Angular Accelerationα

## Difference between Linear SHM and Angular SHM

There are some key differences between Linear and Angular SHM, some of which is as follows:

Linear SHM | Angular SHM |
---|---|

In Linear SHM, the particle moves back and forth along a straight line with constant amplitude and frequency. | In angular SHM, the particle moves in a circular path around an axis with a constant amplitude and frequency. |

Restoring Force is directly proportional to the linear displacement. | Restoring torque is directly proportional to the angular displacement. |

Some examples of Linear SHM include a spring-mass system, a piston in the automobile, etc. | Some examples of Angular SHM include the motion of a pendulum, a rotating fan, or a balance wheel in a watch. |

**Equations for Simple Harmonic Motion**

**Equations for Simple Harmonic Motion**

Let’s consider a particle of mass (m) doing Simple Harmonic Motion along a path A’OA the mean position is O. Let the speed of the particle be V_{0} when it is at position P (at some distance from point O)

At the time, ** t = 0 **the particle at P (moving towards point A)

At the time, ** t = t** the particle is at Q (at a distance X from point O) at this point if velocity is V then:

The force F acting on a particle at point p is given as,

** F = -K X **[where, K = positive constant]

We know that,

** F = m a **[where, a = Acceleration at Q]

⇒ m a = -K x

⇒ a = -(K/m) x

As K/m = ω^{2}

Thus, a = -ω^{2}x

Also, we know a = d^{2}X/d^{2}t]

Therefore, d^{2}x/d^{2}t = -ω^{2}x

d^{2}x/d^{2}t + ω^{2}x = 0which is the differential equation for linear simple harmonic motion.

## Solutions of Differential Equations of SHM

The solutions to the differential equation for simple harmonic motion are as follows:

Equation of SHM is, d

^{2}x/d^{2}t + ω^{2}x = 0Multiply by 2 \frac{dx}{dt}, to get

After integration, we get a separable equation

Integrating,

This is the required Solution of the SHM Equation.

### Different Cases of the Solution of SHM Equation

For particle is in its mean position at point (O) [ϕ =0], displacement function becomes

x = Asinωt.For t = 0, when object is at rest, displacement function becomes

x = AsinϕFor particle in any position throughout the SHM (any time t), displacement function becomes

x = Asin(ωt+ϕ)

**Energy in Simple Harmonic Motion (SHM)**

**Energy in Simple Harmonic Motion (SHM)**

A system performing SHM is called a Harmonic Oscillator. The energy of the particle performing the SHM is discussed below in the particle.

Let’s take a particle of mass (** m**) performing linear SHM with angular frequency (

**)**

**ω****and the amplitude of the particle is (A)**

Now we know that the displacement of the particle at any time is given using the SHM equation,

x = A sin (ωt + Φ)…(i)

where ** Φ** is the phase difference.

Differentiating eq(i) wrt time we get,

v = Aω.cos (ωt + Φ)

v = ω. Acos (ωt + Φ)

v = ω.√(A^{2} – x^{2})…(ii)

Again, differentiating eq(ii) wrt time we get,

a = -ω^{2}. Asin (ωt + Φ)

a = -ω^{2}x

Restoring force acting on the body is,

F = -kx

where, k = mω^{2}

Now for the energy of the SHM particle.

### Kinetic Energy of Particles in SHM

Kinetic Energy(K.E) = 1/2 mv^{2} {v^{2} = ω^{2}(A^{2} – x^{2})}

K.E = 1/2 mω^{2}(A^{2}– x^{2})

Also, the kinetic energy of the particle in SHM is,

K.E = 1/2 mω^{2}A^{2}cos^{2}(ωt + Φ)

### Potential Energy of Particles in SHM

For the potential energy we know that,

Potential Energy(P.E) = – Work Done

P.E = -F.dx

P.E = kxdx (As dx is also negative)

Integrating from o to x

P.E = (kx^{2})/2

We know that, k = mω^{2}

P.E = (mω^{2}x^{2})/2

We know that, {x = Asin(ωt + Φ)}

P.E = (mω^{2})/2. A^{2}sin^{2}(ωt + Φ)

### Total Mechanical Energy of the Particle in SHM

Total Energy(E) = Kinetic Energy(K.E) + Potential Energy(P.E)

E = 1/2 mω^{2}(A^{2} – x^{2}) + (mω^{2}x^{2})/2

E = 1/2 mω^{2}A^{2}

This is the total Energy of the particle in SHM.

## Simple Harmonic, Periodic, and Oscillation Motion

It seems that Simple Harmonic, Periodic, and Oscillation Motions are the same but they are indeed different. Now let’s learn about them in detail.

### Simple Harmonic Motion

The motion of an object around a mean position in which the acceleration of the particle is directly proportional to the displacement of the particle is called Simple Harmonic Motion. Such as the motion of a cantilever. We can say that all Simple Harmonic Motions are oscillatory and periodic, but the converse is not true.

### Periodic Motion

Periodic motion is defined as the motion of any object that repeats its motion after a fixed interval of time. Such as the motion of the Moon around the Earth.

### Oscillation Motion

Oscillatory motion is the to-and-fro motion of an object from its mean position. SHM is an example of Oscillatory motion.

## Difference between Periodic, Oscillation, and Simple Harmonic Motion

There are some differences between Periodic motion, Oscillatory Motion, and Simple Harmonic Motion, which are listed as follows:

Periodic Motion | Oscillatory Motion | Simple Harmonic Motion |
---|---|---|

The motion of an object is said to be periodic if it moves in such a way that it repeats its path at regular intervals of time. | The to-and-fro motion of an object from its mean position is defined as oscillatory motion. | Simple Harmonic Motion is the motion of an object, around a mean position in which the acceleration of the particle is directly proportional to the displacement of the particle. |

Some Examples of Periodic motion include the motion of hands in a clock, the movement of the Earth around the Sun, and a simple pendulum. | Examples of oscillatory motion include a simple pendulum, a vibrating tuning fork, AC current, waves such as sound waves, ocean waves, or light waves, etc. | Examples of SHM are the motion of a spring system i.e., a mass attached to the end of a spring, swings, and the motion of a cantilever. |

Periodic motion is not always harmonic. | Oscillatory Motion is not always periodic, | SHM is an example of Periodic and Oscillatory motion. |

**Read More,**

**Sample Questions on Simple Harmonic Motion**

**Sample Questions on Simple Harmonic Motion**

**Question 1: Why is Harmonic Motion Periodic?**

**Solution: **

The sine wave can represent a harmonic motion. When a spring is stretched from its mean position, it oscillates to and fro about the mean position under the influence of a restoring force that is always directed towards the mean position and whose magnitude at any instant is proportional to the body’s displacement from the mean position at that instant. When there is no friction, the motion tends to be periodic. The harmonic motion is periodic in this case.

**Question 2: What are Periodic and Non-Periodic Changes?**

**Solution: **

Periodic changes are those that occur at regular intervals of time, such as the occurrence of day and night, or the change of periods in your school. Non-periodic changes are those that do not occur on a regular basis, such as the freezing of ice to water.

**Question 3: What is the period of the Earth’s revolution around the sun and around its polar axis? what is the motion Earth performs explain?**

**Solution: **

The earth’s revolution around the sun takes one year, and its revolution around its polar axis takes one day. The motion of earth is periodic because after some interval of time it repeats its path.

**Question 4: What is the frequency of SHM? How time periods and frequency are related?**

**Solution: **

The frequency of SHM is the number of oscillations performed by a particle per unit of time. Hertz, or r.p.s. (rotations per second), is the SI unit of frequency. Frequency and time period are related as:

Frequency, (f) = 1/ Time period (T)

**Question 5: A spring with a spring constant of 1200 N m–1 is mounted on a horizontal table. A 3 kg mass is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 m before being released. Determine the following:**

**The frequency of oscillations,****Maximum acceleration of the mass, and****The maximum speed of the mass.**

**Solution:**

Given:

- Spring Constant, k = 1200 N/m.
- Mass of Object, m = 3 kg.
- Displacement, x = 2 m.

(1) Frequency of Oscillation:We know that frequency (f) = 1/Time period (T) T = 2π/ω and ω = √k/m]

Therefore,

f = (1/2π)√k/m

= (1/2 × 3,14) √1200/3 = 3.18 Hz.

(2)Maximum Acceleration:Maximum Acceleration (a) = ω

^{2}xwhere, ω = Angular frequency = √k/m

Therefore, a = x(k/m)

a = 2 × (1200/3)

a= 800 m/s

^{2}.

(3) Maximum Speed:Maximum Speed (V) = ωx

Put, ω = √k/m.

Therefore, V = x(√k/m)

V = 2 × (√1200/3)

V = 40 m/s.

**Question 6: A mass of 2 kg is attached to the end of the spring with a spring constant of 50 N/m. What is the period of the resulting simple harmonic motion? (π = 3.14)**

**Solution:**

Formula for time period is

T = 2π√(k/m)where,

is the massmis the spring constantkThus, T = 2π√(50/2)

⇒ T = 2π√(25)

⇒ T = 2π/5

⇒ T ≈ 1.26 s

So, the time period of the SHM is approximately 1.26 s.

**Question 7: A block of mass 0.5 kg is attached to the end of the spring (spring constant =100 N/m). If The block is displaced 0.1 m from its equilibrium position then what is the maximum speed of the block during its motion?**

**Solution:**

The maximum speed of the block is given by:

v_{max}= Aωwhere,

is Amplitude of MotionAis Angular FrequencyωAlso, angular frequency ω is given by:

ω = √(k/m)where,

is the massmis the spring constantkGiven:

- Amplitude(A) = 0.1 m
- k = 100 N/m
- m = 0.5 Kg
⇒ v

_{max}= 0.1 × √(100/0.5)⇒ v

_{max}= 0.1 × √(1000/5)⇒ v

_{max}= 0.1 × √(200)⇒ v

_{max}= √2

So, the maximum speed of the block during its motion is √2 m/s.

## FAQs on Simple Harmonic Motion

### 1. What is Simple Harmonic Motion?

Simple Harmonic Motion is the Periodic and Oscillatory motion, where a particle moves around an mean position and the acceleration of the particle is directly proportional to the displacement of the particle.

### 2. What are Examples of Simple Harmonic Motion?

A spring-mass system (a block of mass attached to the end of the spring), a simple pendulum, a vibrating tuning fork, etc. are some examples of Simple Harmonic Motion.

### 3. What is Equation for Simple Harmonic Motion?

The equation for simple harmonic motion is given as:

x = A cos(ωt + φ)

Where

xis the Displacement of the Particle from its Equilibrium Position at TimetAis the Amplitudeωis the Angular Frequencyφis the Initial Phase Angle

### 4. What is Time Period of Simple Harmonic Motion?

The time taken by the particle to complete one oscillation is the period of the SHM, and it is given by:

T = 2π/ω

whereωis the angular frequency of the motion.

### 5. What is Resonance in Simple Harmonic Motion?

Resonance is the phenomenon, when the frequency of external driving force matches the natural frequency of the system, causing the amplitude of the motion to increase significantly. In SHM, resonance can cause the amplitude of the oscillation to become significantly large, which can cause potential damage to the system.

### 6. Is it Possible for a Motion to be Oscillatory but not Simple Harmonic?

When a ball is dropped from a height onto a perfectly elastic surface, the motion of the ball is oscillatory but not simple harmonic because the restoring force, F = mg = constant.

### 7. What is Angular Frequency in Simple Harmonic Motion?

Angular frequency (ω), also known as radial or circular frequency, is a unit of time for measuring angular displacement. As a result, its units are degrees (or radians) per second.

The angular frequency(ω) is given by the expression,

**ω = 2π/T**

**SHM JEE Mains Questions**

**SHM JEE Mains Questions**

**1. A damped harmonic oscillator has a frequency of 5 oscillations per second. For every 10 oscillations, the amplitude of the oscillator drops to half. Find the time taken to drop the amplitude to 1/1000 of the original value.**

**2. If the length of a simple pendulum in SHM is increased by 21% then what is the percentage increase in the time period of the pendulum of the increased length**

**3. It is given that the ratio of maximum acceleration to maximum velocity in a SHM is 10 second**^{-1}** and at t = 0, the displacement is 5 m. What is the maximum acceleartion? Given that the initial phase is π/4**

**4. If a child is swinging in a sitting position and then he stands up, then how the time period of the swing will be affected.**

**5. The displacement of a particle in simple harmonic motion is given b y x(t) = Asin(πt/90). Find the ratio of kinetic energy to the potential energy at t = 210 seconds**