Simple Pendulum

A mechanical system in which oscillatory motion takes place is known as the simple pendulum. In this system, a point mass ‘m’ is present which is suspended by means of long thread that is inextensible with a string of length ‘L’ from a fixed rigid support. The gravitational force drives the motion in a simple pendulum and it occurs in a vertical plane. Here mg sin θ is the tangential component of the gravitational force, which is always acted towards the mean position where θ=0, that is, opposite to the displacement, and a restoring force is acted upon the arc.

Simple Pendulum- Relationship between Time Period and Frequency

\[F_{t} = -mg sin \theta = \frac{md^{2}s}{dt^{2}}\]

Now, in the case of small-displacement where \[sin \theta \equiv 0\] and along the arc is the motion of the bob, then

s=Lθ

Therefore,

\[ \frac{d^{2} \theta}{dt^{2}} = - \frac{g}{L}sin \theta = - \frac{g}{L} \theta \]

or

\[ \frac{d^{2}\theta}{dt^{2}} + \frac{g}{L} \theta = 0 \]

\[ \frac{d^{2} \theta}{dt^{2}} + \omega_{n}^{2} \theta = 0\]

where \[\omega_{n}\] is given by 

\[\omega_{n} = \sqrt{\frac{g}{L}}\]

and the period of motion is

\[T = \frac{2\pi}{\omega_{n}} = 2\pi \sqrt{\frac{L}{g}}\]

\[\theta = A sin (\omega_{n} t + \phi)\]

This equation depicts that the only factors on which the period and frequency of a simple pendulum depend are the value of g and the length of the string.

Simple Pendulum- Total Energy

Kinetic Energy

The energy which is possessed by an object because of the motion in it is known as the kinetic energy of that object. It depends upon both; mass and speed. In case of a simple pendulum, the kinetic energy is increased as the equilibrium position is reached by the pendulum while on the other hand if the pendulum bob has moved further away from the equilibrium position the kinetic energy will decrease. 

The kinetic energy is given by-

\[\text{Kinetic Energy} = \frac{1}{2}. m. v^{2}\]

Potential Energy

The energy which is possessed by an object when it is in the resting phase is known as the potential energy of that object. The type of potential energy which is possessed by a simple pendulum is known as the gravitational potential energy, which is dependent upon the mass ‘m’ of the object, height ‘h’, and a gravitational force ‘g’. In the case of simple pendulum, the lowest position is referred to as the zero level or the reference position. Therefore, the height and the potential energy is 0 when the bob is at its equilibrium position while when the pendulum moves back and forth the pendulum moves further away from the equilibrium position and the height gets increased as it moves. When the height reaches the maximum point, the maximum displacement is reached from the equilibrium position. Then again, when the bob moves towards the equilibrium position the height gets decreased and so does its potential energy. The potential energy is given by-

\[\text{Potential Energy = m. g. h} \]

Total Mechanical Energy

The total amount of energy that is possessed by the pendulum is known as the Total Mechanical Energy which is simply the sum of the two forms of energy of the pendulum,i.e.,  Kinetic Energy and Potential Energy.

\[\text{Total Mechanical Energy = Kinetic Energy + Potential Energy}\]

\[\text{Total Mechanical Energy} = \frac{1}{2}.m.v^{2} + m.g.h\]

Simple Pendulum- Solved Example

Example-  If T= 1.2 s of a simple pendulum in a place where g= 9.8 m/s2. What is the length of the pendulum?

Solution- Given 

T= 1.2 s

g=9.8 m/s2

l=?

We know that

\[T = 2\pi \sqrt{\frac{L}{g}}\]

Therefore

\[T = 4\pi^{2} \times \frac{l}{g}\]

\[l = \frac{T^{2}g}{4\pi^{2}} = \frac{(1.2)^{2} (9.8)}{4(3.142)^{2}}\]

\[ l= 0.36 m \]

More about the Simple Pendulum

To simply define simple pendulum, it is a bob that comprises of a molecule of mass ‘m’ suspended by a massless non-expandable string of length ‘L’ in an area of space in which there is a consistent, uniform gravitational field, e.g. near the surface of the earth. The suspended molecule is called the pendulum bob. 

Thus, a pendulum is typically a weight that is hung from a fixed point. It is positioned in such a way that it enables the device to swing freely back and forth. Further, the string from which a pendulum bob is hanging is of negligible mass. Refer to the simple pendulum diagram below:

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Pendulum Formula

A mass ‘m’ hung by a string of length ‘L’ is a simple pendulum and undergoes simple harmonic motion for amplitudes approximately below 15º. The time period of simple pendulum derivation is T = 2π√Lg T = 2 π L g, where

‘L’ = the length of the string

T = Time period in seconds

‘g’ = the acceleration owing to gravity (9.8 m/s² on Earth).

π = Pi (values 3.14)

Important Terms Associated with Simple Pendulum

1. Oscillation of Simple Pendulum: When the spherical bob of the pendulum supplants at the primary angle and then liberated, the pendulum is observed to move in a repeated back-and-forth motion periodically. This motion is called the oscillatory motion. The central point of oscillation is termed as an equilibrium position.

2. The Time Period of Simple Pendulum Motion: The time taken by the pendulum to complete one full oscillation is termed as the time period of oscillation and is symbolically represented by the letter "T".

3. Length of the Simple Pendulum: The distance between the point of suspension to the center of the bob is the length of a pendulum and is denoted by the symbol "l".

4. The amplitude of Simple Pendulum Motion: The distance traveled by the pendulum from the point of equilibrium position to one side is referred to the amplitude of oscillation of the simple pendulum.

The Period of Oscillation of a Simple Pendulum

If you pull the pendulum bob to one side and let it go, you find that it swings to and fro, meaning that it oscillates. At this point, you are not aware of or not the bob goes through simple harmonic motion, but you know that it oscillates. 

To identify whether it undergoes simple harmonic motion, all you have to do is to establish whether its acceleration is a negative constant times its position. Since the bob moves on an arc rather than a line, it is easier to assess the motion using angular variables. 

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Solved Example on Pendulum

Example: A pendulum has a length of 4 meters. It accomplishes one full cycle of 0.25 times every second. The amplitude that the pendulum bob reaches from the centre is 0.1 meters. Calculate the time period of the oscillation? What is the displacement after 0.6 seconds?

Here,

Length of the pendulum (L) = 4 m

Frequency of the pendulum = 0.25

Amplitude or maximum displacement= 0.1

Time = is 0.6

Acceleration due to gravity (g), as always (g=9.8).

In order to find out T, we use the time period of simple pendulum formula i.e.

T = 2π√Lg

Thus, we get

2π0.4082

2π × 0.64

2×3.14 × 0.64 = 4.01

Hence, the time period of the oscillation is 4.01 seconds.

Fun Facts

A pendulum is one of the frequent items found in households, which is commonly observed in wall clocks.

The time period of a simple pendulum doubles when the length of the pendulum increases by four times. This depicts a square root relation between the period and length of a simple pendulum.

Conclusion 

Vedantu has focused its energy on making sure that students grasp the intensity of concepts and solutions to questions in an elaborate manner. You can find everything you’re looking for at the website in the form of a PDF File. These PDF files are also downloadable for free, students can go through the concepts and Definitions of the Simple Pendulum, the relationship between Time period and Frequency, and the total energy while solving questions to understand carefully the concepts used to solve these questions. This will be of great help to the students for both school and competitive exams.