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For small angle, we can consider " sin θ θ " as a good approximation. Hence,

α = - m g L I θ

We have just seen the condition that results from the requirement of SHM. This condition requires that angular amplitude of oscillation should be a small angle.

Angular frequency

Comparing the equation obtained for angular acceleration with that of “ α = - ω 2 θ ”, we have :

ω = m g L I

There is yet another aspect about moment of inertia that we need to discuss. Note that we have considered that bob is a point mass. In that case,

I = m L 2

and

ω = m g L m L 2 = g L

We see that angular frequency is independent of mass. What happens if bob is not a point mass as in the case of real pendulum. In that case, angular frequency and other quantities dependent on angular frequency will be dependent on the MI of the bob – i.e. on shape, size, mass distribution etc.

We should understand that requirement of point mass arises due to the requirement of mass independent frequency of simple pendulum – not due to the requirement of SHM. In the nutshell, we summarize the requirement of simple pendulum that arises either due to the requirement of SHM or due to the requirement of mass independent frequency as :

  • The pivot is free of any energy loss due to friction.
  • The string is un-strechable and mass-less.
  • There is no other force (other than gravity) due to external agency.
  • The angular amplitude is small.
  • The ratio of length and dimension of bob should be large so that bob is approximated as point.

Time period and frequency

Time period of simple pendulum is obtained by applying defining equation as :

T = 2 π ω = 2 π L g

Frequency of simple pendulum is obtained by apply defining equation as :

ν = 1 T = 1 2 π g L

Special cases of simple pendulum

We have so far discussed a standard set up for the study of simple pendulum. In this section, we shall discuss certain special circumstances of simple pendulum. For example, we may be required to analyze motion of simple pendulum in accelerated frame of reference or we may be required to incorporate the effect of change in the length of simple pendulum.

Second pendulum

A simple pendulum having time period of 2 second is called “second” pendulum. It is intuitive to analyze why it is 2 second - not 1 second. In pendulum watch, the pendulum is the driver of second hand. It drives second hand once (increasing the reading by 1 second) for every swing. Since there are two swings in one cycle, the time period of second pendulum is 2 seconds.

Simple pendulum in accelerated frame

The time period of simple pendulum is affected by the acceleration of the frame of reference containing simple pendulum. We can carry out elaborate force or torque analysis in each case to determine time period of pendulum. However, we find that there is an easier way to deal with such situation. The analysis reveals that time period is governed by the “effective” acceleration or the “relative” acceleration given as :

g = g a

where g’ is effective acceleration and “ a ” is acceleration of frame of reference (a≤g). We can evaluate this vector relation for different situations.

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Oscillation and wave motion. OpenStax CNX. Apr 19, 2008 Download for free at http://cnx.org/content/col10493/1.12
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