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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

Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
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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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