Simple Pendulum

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What is a 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:

A. 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.


B. 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".


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


D. Amplitude of Simple Pendulum Motion: The distance travelled 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 double 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.

FAQ (Frequently Asked Questions)

Q1: What is the Time Period of a Simple Pendulum?

Ans: The time period of a simple pendulum is the time taken by the pendulum bob to complete one full oscillation. The time period of simple pendulum derivation is given as

T= 2π √ (L/g)

Q2: What is the Actual Time Period of a Simple Pendulum?

Ans: A simple pendulum abides by the following equation of motion,

d2θ dt2 + gL sin (θ) = 0.

The solution to this equation has a period stated by:-

T=4L2g−−√∫θ001cos (θ) − cos(θ0) dθ

This is what we mean by “actual” period.

Q3. What is the Time Period of a Pendulum with Infinite Length?

Ans: You already know the formula of the time period of a pendulum i.e. T=2π√ l/g. However, the given formula is only applicable if the length of the pendulum bob is very short while in comparison to the radius of earth or on any other planet with its radius (r) and (g) value.

The formula T=2π√[l/g] is derived from the formula i.e. T=2π 1/g{ 1/L + 1/R}√.

Now with this formula, if we put the value of R=6400000m then 1/6400000 emerges to be negligible and it will give us T=2π√ [l/g].

On the other hand, when the length of the pendulum is large enough to compare with R then you have to consider R while making calculations.

If the length of the pendulum is expanded to infinity then 1/l will come out as 1/∞ = 0 and the formula will render:-

T=2π√ {R/g} ≈ 84.5 minutes

What’s even more amazing to know is that the time period of a satellite orbiting near the surface of the earth is also 84.5 minutes. When you see a horizontal circular motion in a horizontal (parallel) view, it will appear like a simple harmonic motion (SHM) but when you look at it from the top it will appear like a circular motion.