Important Study Notes on Simple Pendulum

Introduction

Oscillation is defined as the movement of back and forth in a regular rhythm. It is repetitive, or you can say a periodic variation. It is a measure between two or more different states or a periodic variation about its central value. In other words, vibration is usually used to describe a mechanical oscillation. It’s the equilibrium value in time. Some of its common examples are simple pendulum variations in an altering current.

Astronomer Christiaan Huygens became the first to report the coupled oscillation phenomenon in two pendulum clocks.

Simple Harmonic Motion

Simple Harmonic Motion or SHM is a motion in which the object moves in a to and fro motion along a line. It is a name that is given to oscillatory motion. In this motion, the acceleration of the particle is directly proportional to its displacement. It is directed towards its mean position. The simple harmonic motion is periodic and can be represented by a single harmonic function like sine or cosine. This motion forms a base for the more difficult period motion, which is calculated through a technique known as Fourier Analysis. It is through the methods of Fourier analysis. The particle’s displacement is measured in terms of linear displacement.

What is a Simple Pendulum?

Simple Pendulums are defined as a device where its point mass is attached to a light inextensible string and is suspended from a fixed support. It is an arrangement of mechanics with a small mass, also called pendulum bob of a mass, let’s say ‘m’, balanced by a slim but strong string attached to a stage on the upper having a length L. The length of this simple pendulum is denoted as the vertical gap between the point from which it is suspended and the centre of mass (COM) of the suspended body. Pendulums have a common usage that is witnessed in several instances like a clock to keep track of time, in an uncommon way such as a sinker near a fishing line. Hence, any ideal pendulum with a mass of (m) suspended by a flexible but inextensible thread is kept free from vibrations. 

Derivation of Simple Pendulums

Period of Simple Pendulums Derivation

Before this, it is important to know the time period (T) of any simple pendulum and the time taken by the pendulum to complete one full oscillation. It is symbolised by the letter ‘T’.

If we use the motion’s equation, T – mg cosθ = mv 2 L

The torque tends to bring the given mass back to its position of equilibrium,

τ = mgL × sinθ = mgsinθ × L = I × α

For smaller oscillation angles, sin θ ≈ θ,

Then, Iα = -mgLθ

α = -(mgLθ)/I

– ω02 θ = -(mgLθ)/I

ω02 = (mgL)/I

ω0 = √(mgL/I)

If we use I = mL2, [I – MOI of the Bob]

we will get, ω0 = √(g/L

Hence, the (T) time period of any simple pendulum can be written as,

T = 2π/ω0 = 2π × √(L/g)

Hence, it is the Simple Pendulum Derivation

Difference between Physical and Simple Pendulum

There are several bases on which it is differentiated:

• Model

A simple Pendulum is an ideal model as sometimes it is difficult to achieve in reality. In contrast, the physical pendulum is considered a realistic model of the simple pendulum and also has a finite body and shape.

• Tension

In the simple pendulum, a tension force acts on the string, which eventually helps the object suspend. The physical pendulum doesn’t need any string for suspension, so there is no tension.

• Oscillation

Concerning oscillation, the Simple Pendulum oscillates in a short time interval with a large angle, whereas in the physical pendulum, the angle of oscillation is small.

• Gravity

In simple pendulums, gravity acts at the centre of the pendulum’s bob, whereas gravity acts towards the centre of mass in the physical pendulum.

• Suspension

A string is needed to suspend from rigid support in the Simple Pendulum, whereas no string is needed for the suspension in the physical pendulum.

Conclusion

So, to conclude here, the first observation made by Galileo tells that the time taken by a pendulum to swing back and forth through the small distances depends upon the length of the pendulum only. The Simple Pendulum observation, the differences with that of a physical pendulum and the Simple Pendulum observation is the very important part that is covered to give you a wholesome view of its workings and facts.