# 1.4 The simple pendulum

 Page 1 / 3
• Measure acceleration due to gravity.

Pendulums are in common usage. Some have crucial uses, such as in clocks; some are for fun, such as a child’s swing; and some are just there, such as the sinker on a fishing line. For small displacements, a pendulum is a simple harmonic oscillator. A simple pendulum    is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in [link] . Exploring the simple pendulum a bit further, we can discover the conditions under which it performs simple harmonic motion, and we can derive an interesting expression for its period.

We begin by defining the displacement to be the arc length $s$ . We see from [link] that the net force on the bob is tangent to the arc and equals $-\text{mg}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$ . (The weight $\text{mg}$ has components $\text{mg}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta$ along the string and $\text{mg}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$ tangent to the arc.) Tension in the string exactly cancels the component $\phantom{\rule{0.25em}{0ex}}\text{mg}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta$ parallel to the string. This leaves a net restoring force back toward the equilibrium position at $\theta =0$ .

Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator. In trying to determine if we have a simple harmonic oscillator, we should note that for small angles (less than about $\text{15º}$ ), $\text{sin}\phantom{\rule{0.25em}{0ex}}\theta \approx \phantom{\rule{0.25em}{0ex}}\theta \phantom{\rule{0.25em}{0ex}}$ ( $\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$ and $\theta$ differ by about 1% or less at smaller angles). Thus, for angles less than about $\text{15º}$ , the restoring force $F$ is

$F\approx -\text{mg}\mathrm{\theta .}$

The displacement $s$ is directly proportional to $\theta$ . When $\theta$ is expressed in radians, the arc length in a circle is related to its radius ( $L$ in this instance) by:

$s=\mathrm{L\theta },$

so that

$\theta =\frac{s}{L}.$

For small angles, then, the expression for the restoring force is:

$F\approx -\frac{\text{mg}}{L}s$

This expression is of the form:

$F=-\text{kx},$

where the force constant is given by $k=\text{mg}/L$ and the displacement is given by $x=s$ . For angles less than about $\text{15º}$ , the restoring force is directly proportional to the displacement, and the simple pendulum is a simple harmonic oscillator.

Using this equation, we can find the period of a pendulum for amplitudes less than about $\text{15º}$ . For the simple pendulum:

$T=2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{m}{\text{mg}/L}}.$

Thus,

$T=2\pi \sqrt{\frac{L}{g}}$

for the period of a simple pendulum. This result is interesting because of its simplicity. The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass. As with simple harmonic oscillators, the period $T$ for a pendulum is nearly independent of amplitude, especially if $\theta$ is less than about $\text{15º}$ . Even simple pendulum clocks can be finely adjusted and accurate.

Note the dependence of $T$ on $g$ . If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity. Consider the following example.

what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
Got questions? Join the online conversation and get instant answers!