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  • Measure acceleration due to gravity.
In the figure, a horizontal bar is drawn. A perpendicular dotted line from the middle of the bar, depicting the equilibrium of pendulum, is drawn downward. A string of length L is tied to the bar at the equilibrium point. A circular bob of mass m is tied to the end of the string which is at a distance s from the equilibrium. The string is at an angle of theta with the equilibrium at the bar. A red arrow showing the time T of the oscillation of the mob is shown along the string line toward the bar. An arrow from the bob toward the equilibrium shows its restoring force asm g sine theta. A perpendicular arrow from the bob toward the ground depicts its mass as W equals to mg, and this arrow is at an angle theta with downward direction of string.
A simple pendulum has a small-diameter bob and a string that has a very small mass but is strong enough not to stretch appreciably. The linear displacement from equilibrium is s size 12{s} {} , the length of the arc. Also shown are the forces on the bob, which result in a net force of mg sin θ size 12{ - ital "mg""sin"θ} {} toward the equilibrium position—that is, a restoring force.

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 size 12{s} {} . We see from [link] that the net force on the bob is tangent to the arc and equals mg sin θ size 12{ - ital "mg""sin"θ} {} . (The weight mg size 12{ ital "mg"} {} has components mg cos θ size 12{ ital "mg""cos"θ} {} along the string and mg sin θ size 12{ ital "mg""sin"θ} {} tangent to the arc.) Tension in the string exactly cancels the component mg cos θ size 12{ ital "mg""cos"θ} {} parallel to the string. This leaves a net restoring force back toward the equilibrium position at θ = 0 size 12{θ=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 15º size 12{"15"°} {} ), sin θ θ size 12{"sin"θ approx θ} {} ( sin θ size 12{"sin"θ} {} and θ size 12{θ} {} differ by about 1% or less at smaller angles). Thus, for angles less than about 15º size 12{"15"°} {} , the restoring force F size 12{F} {} is

F mg θ. size 12{F= - ital "mg"θ} {}

The displacement s size 12{s} {} is directly proportional to θ size 12{θ} {} . When θ size 12{θ} {} is expressed in radians, the arc length in a circle is related to its radius ( L size 12{L} {} in this instance) by:

s = , size 12{s=Lθ} {}

so that

θ = s L . size 12{θ= { {s} over {L} } } {}

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

F mg L s size 12{F approx - { { ital "mg"} over {L} } s} {}

This expression is of the form:

F = kx , size 12{F= - ital "kx"} {}

where the force constant is given by k = mg / L and the displacement is given by x = s size 12{x=s} {} . For angles less than about 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 15º . For the simple pendulum:

T = m k = m mg / L . size 12{T=2π sqrt { { {m} over {k} } } =2π sqrt { { {m} over { ital "mg"/L} } } } {}


T = L g size 12{T=2π sqrt { { {L} over {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 size 12{T} {} for a pendulum is nearly independent of amplitude, especially if θ size 12{θ} {} is less than about 15º size 12{"15"°} {} . Even simple pendulum clocks can be finely adjusted and accurate.

Note the dependence of T size 12{T} {} on g size 12{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.

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Source:  OpenStax, General physics ii phy2202ca. OpenStax CNX. Jul 05, 2013 Download for free at http://legacy.cnx.org/content/col11538/1.2
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