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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} } } } {}

Thus,

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

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
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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
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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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