In view of the coronavirus pandemic, we are making LIVE CLASSES and VIDEO CLASSES completely FREE to prevent interruption in studies
Physics > Oscillations > Some Systems Executing Simple Harmonic Motion
Oscillations

Some Systems Executing Simple Harmonic Motion

We have already studied Simple Harmonic Motion (S.H.M.). But do systems execute purely simple harmonic motion in practice? No, but under certain conditions, they execute simple harmonic motion approximately. For example, a simple pendulum. Let’s learn about some of these systems and their harmonic motion.

Suggested Videos

Play
Play
Play
Arrow
Arrow
ArrowArrow
Introduction to Periodic Motion H
SHM as a projection of circular motion I H
VST Simple Harmonic Motion Problem 4 and its Solution
Slider

 

Systems executing S.H.M.

We are going to study about two systems below that perform Simple Harmonic Motion. These systems include- a spring-block system and a simple pendulum. Let’s understand one by one how these systems perform simple harmonic motion and under what conditions.

Browse more Topics under Oscillations

Download the Cheat Sheet of Oscillations below

Oscillations due to a Spring

pendulum

The small oscillations of a block fixed to a spring, which in turn is fixed to a wall are the simplest example of simple harmonic motion. Let the mass of the block be ‘m’. The block is placed on a frictionless horizontal surface. When we pull the block and then release it, it performs to and fro motion about its mean position. Let x=0 be the position of the block when it is at its equilibrium position. Now,

  • If we pull the block outwards, there is a force exerted by the string that is directed towards the equilibrium position.
  • If we push the block inwards, there is a force exerted by the string towards the equilibrium position.

pendulum

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. Let the restoring force be F and the displacement of the string from the equilibrium position be x.

Therefore, we can deduce from this system that the magnitude of restoring force exerted by a system is directly proportional to the displacement of the system from its equilibrium position.The restoring force always acts in the opposite direction of that of displacement. Therefore,

F= -kx

Here, k is the constant which is called the force constant. In this system, it is called the spring system. The value of k depends on the stiffness of the spring. A stiff spring will have larger k and a soft spring will have small k. You can see that this equation is the same as the Force law of Simple Harmonic Motion. Therefore, a spring system executes simple harmonic motion. From equation I, we have, ω= √k/m

∴ The time period (T) of the oscillator = 2π√m/k

The Simple Pendulum

A practical simple pendulum is a small heavy sphere(bob) suspended by a light and inextensible string from a rigid support. The length of the simple pendulum is the distance between the point of suspension and centre of gravity of heavy sphere.

The Simple Harmonic Motion Pendulum

The motion of Simple Pendulum as Simple Harmonic Motion

When we pull a simple pendulum from its equilibrium position and then release it, it swings in a vertical plane under the influence of gravity. It begins to oscillate about its mean position. Therefore, the motion is periodic and oscillatory. Now, if we displace the pendulum by a very small angle Θ, then it performs the simple harmonic motion.

Simple Pendulum

Consider a simple pendulum having mass ‘m’, length L and displaced by a small angle Θ with the vertical. Thus, it oscillates about its mean position. In the displaced position, two forces are acting on the bob,

  • Gravitational force, which is the weight of the bob – ‘mg’ acting in the downward direction.
  • Tension T’ in the string.

Now, weight mg is resolved in two components,

  • Radial component mgcosΘ along the string.
  • Tangential component mgsinΘ perpendicular to the string.

Radial component mgcosΘ is balanced by Tension T’ in the string and the Tangential component mgsinΘ is the restoring force acting on mass m towards the equilibrium position. Therefore,

Restoring force, F =-mgsinΘ

The negative sign indicates that F and Θ are in opposite directions. Note that the restoring force is proportional to sinΘ instead of Θ. Therefore, it is not yet a simple harmonic motion. However, if the angle is very small, we can assume that sinΘ is nearly equal to Θ in radian. Thus, the displacement x= LΘ and for a small angle, it is nearly a straight line.

∴ Θ= x/L

Hence, assuming sinΘ= Θ, F = -mgΘ = -mgx/L. As m,g, and L are constant, F α -x. For a small displacement, the restoring force is directly proportional to the displacement and its direction is opposite to that of displacement. Therefore, Simple pendulum performs linear S.H.M.

As we know, acceleration= force/mass = -mgx/L/m
∴ acceleration= -gx/L
∴ acceleration per unti displacement will be,
acceleration/x = -g/L
Also, acceleration = ω2x
∴ ω = √acceleration per unit displacement
Considering magnitude, acceleration/x = g/L = ω2

Period of a Simple Pendulum

Now, the period of a simple pendulum is, T = 2π/ω = 2π/√acceleration per unit displacement

∴ T = 2π/√g/L = 2π√L/g

The above equation is the equation of period of a simple pendulum

The frequency of  Simple Pendulum

Now, frequency of a simple pendulum = f = 1/T = 1/2π√g/L. This equation is the equation of frequency of a simple pendulum.

Solved Question for You on Simple Pendulum Equations

Q: When does a simple pendulum perform Linear Simple Harmonic Motion?

Solution: In an ideal situation, a simple pendulum will execute periodic oscillation with constant amplitude. But in practice, the amplitude of oscillation will gradually decrease until the body comes to rest. The restoring force is proportional to sinΘ and not Θ(Θ is the angular displacement).

Therefore, this motion is not S.H.M. But for Θ as large as 10°,sinΘ can be approximated by Θ to an error in fourth decimal point. For Θ less than 6°, sinΘ ≈ Θ. Thus, for small displacement, the restoring force is directly proportional to the displacement and the pendulum performs linear S.H.M.

Share with friends

Customize your course in 30 seconds

Which class are you in?
5th
6th
7th
8th
9th
10th
11th
12th
Get ready for all-new Live Classes!
Now learn Live with India's best teachers. Join courses with the best schedule and enjoy fun and interactive classes.
tutor
tutor
Ashhar Firdausi
IIT Roorkee
Biology
tutor
tutor
Dr. Nazma Shaik
VTU
Chemistry
tutor
tutor
Gaurav Tiwari
APJAKTU
Physics
Get Started

4
Leave a Reply

avatar
3 Comment threads
1 Thread replies
1 Followers
 
Most reacted comment
Hottest comment thread
4 Comment authors
Tamsin AhmedSubhashMridul Bhaskaranju Recent comment authors
  Subscribe  
newest oldest most voted
Notify of
anju
Guest
anju

what is the expression for instantaneos amplitude of damped harmonic oscillator

Mridul Bhaskar
Guest
Mridul Bhaskar

A=A°*e^(-bt/2m)

Patrick raj
Guest
Patrick raj

Amplitude,A=xe^(-bt/m)

Subhash
Guest
Subhash

How to find angular velocity and damping constant

sdasdas
Guest
sdasdas

you should use antial condition

Tamsin Ahmed
Guest
Tamsin Ahmed

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

Stuck with a

Question Mark?

Have a doubt at 3 am? Our experts are available 24x7. Connect with a tutor instantly and get your concepts cleared in less than 3 steps.
toppr Code

chance to win a

study tour
to ISRO

Download the App

Watch lectures, practise questions and take tests on the go.

Get Question Papers of Last 10 Years

Which class are you in?
No thanks.