Hello dear students we know that in linear simple harmonic motion, particle is acted upon by restoring force towards the mean position and is directly proportional to the displacement in opposite direction. In this article we are going to prove the motion of simple pendulum is also linear SHM. Before that let’s understand what exactly the simple pendulum is! A heavy point mass suspended from the rigid support using inextensible, weightless and twist less spring so as to perform the to and fro motion is called as ideal simple pendulum. But we know that in practice ideal pendulum can not exists.
Let’s define and learn the concept of simple pendulum in detail……..!
Defination:
Laboratory Simple pendulum:–
A heavy point mass suspended from rigid support by a light weight and slightly extensible string is called Laboratory simple pendulum.
Consider the heavy point mass ‘m’ suspended from rigid support using a string of length ‘l’. Let the pendulum is displaced through angle θ from its mean position and oscillates with amplitude ‘x’. As shown in fig below.
When pendulum is at displaced position its weight is resolved into two components as below,
mgcosθ – balances the tension in the string.
mgsinθ -provides restoring force, due to which pendulum is oscillating about its mean position.
Restoring force = -mgsinθ………….(1)
∴ ma = -mgsinθ
∴ acceleration, a = -gsinθ
If θ is very small, then sinθ ≈ θ
∴ acceleration, a = -gθ
But, θ=x/l (∵ length of arc = rθ)
∴ acceleration, a = -g x/l …….(2)
As g and l are constant,
∴acceleration, A α-x ……….(3)
Negative sign indicates that acceleration and displacement are oppositely directed.
From equation (3) it is clear that the acceleration is directly proportional to the displacement. Hence simple pendulum performs linear SHM.
Period of Simple Pendulum:- Time taken by the simple pendulum to complete one oscillation is called as period of simple pendulum.
We know that the period of simple pendulum is given as,
Using above equation we can find the period of simple pendulum.
Some important law related to simple pendulum………!
Law of mass – Period of simple pendulum is independent of mass of pendulum.
Law of length – Period of simple pendulum is directly proportional to square root of length of pendulum.
Law of acceleration due to gravity– Period simple pendulum is inversely Proportional Square to acceleration due to gravity.
Law of isochronous – Period of simple pendulum is independent of amplitude of oscillation.
Different form of pendulum……!
Second’s Pendulum:-
A pendulum whose time period is 2 seconds is called Second’s pendulum.
The time period of simple pendulum is given by,
∴ l = 0.994 m ≈ 1 m = 100 cm.
i.e. when the length of simple pendulum is approximately of 100 cm or 1 m, the pendulum is said to simple pendulum.
Let’s solve some numerical to understand the concept with more details……….!
Ex:-1) Find the period of simple pendulum of length 50 cm oscillating with the amplitude of 12 cm. (use g= 10 m/s2)
Solution:
Here, L= 50 cm=0.5 m, A= 12 cm= 0.12 m
The time period of simple pendulum is given by,
Ex:-2) Find the ratio of periods of simple pendulum of length 50 cm on the surface of earth to that of the surface of planet whose acceleration due to gravity is 1/4th of value of earth.. (use g= 10 m/s2)
Solution:
Here, L= 50 cm=0.5 m, gP= 1/4th of g = g/4
The time period of simple pendulum is given by,
Thanks