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This module is part of a collection of modules intended for students enrolled in a special section of MATH 1508 (PreCalculus) for preengineers. This module addresses the topic of radicals. Radicals play an important role in the modeling of physical phenomena. Several applications of radicals in the field of engineering are presented.

## Introduction

Equations involving radicals abound in the various fields of engineering. Students of engineering must therefore gain confidence and competence in solving equations that include radical expressions. In this module, several different applications that involve the use of radicals to solve engineering problems are presented along with several exercises.

## Centripetal force

Centripetal force is the inward directed force that is exerted on one body as it moves in a circular path about another body.

Figure 1 illustrates a body that is in circular motion about a center point.

As the object moves about the circle, its angle changes. This time rate of change of the angle is called the angular velocity and is denoted by the symbol ω. The angular velocity has units of radians/sec. As an example, if the object makes 2 revolutions in a second, it would have an angular velocity

$\omega =\frac{2\text{revolutions}}{s}=\frac{2\left(2\pi \text{rad}\right)}{s}=4\pi \text{rad}/s$

Examination of Figure 1 shows the centripetal force being directed inward toward the center of the circular path of the object. The velocity of the object is illustrated as being in the direction of the tangent at the point on the circle occupied by the object. If for any reason the body were released from its orbit about the center point, it would travel in a straight line path indicated in the direction of the velocity.

Quite often, one may measure the amount of time that it takes for the object to complete a complete revolution and denote it as the variable ( T ). This value which is usually expressed in seconds is called the period of revolution. For the example given previously where the object makes 2 revolutions per second, the period of revolution ( T ) is ½ second.

The period of revolution ( T ) measured in seconds can be calculated by means of a relationship that involves the magnitude of the centripetal force ( F ) measured in Newtons, the mass of the object ( m ) measured in kilograms, and the radius ( R ) of the circle measured in meters.

$T=\sqrt{\frac{4mR{\pi }^{2}}{F}}$

Question: A mass of 2 kg revolves about an axis. The radius of the object about the axis is 0.5 m. It takes 0.25 seconds for the mass to make a single revolution. What is the value of the centripetal force?

Solution: We begin by replacing the variables of equation (2) by their numeric values

$0\text{.}\text{25}=\sqrt{\frac{4\left(2\right)\left(0\text{.}5\right){\pi }^{2}}{F}}$

Next we take the square of each side of the equation

$\left(0\text{.}\text{25}{\right)}^{2}=\frac{4{\pi }^{2}}{F}$

We can isolate F on the left hand side of the equation as

$F=\frac{4{\pi }^{2}}{0\text{.}\text{625}}$

Which leads to the result $F=\text{632}N\text{.}$

## Nozzle characteristics for aircraft de-icing

The presence of ice on the wings and fuselage on an aircraft can lead to severe problems during stormy winter weather. Equipment is used to spray aircraft with a de-icing agent prior to take-off in order to remove the ice from the wing surfaces and fuselage of planes.

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
how did you get the value of 2000N.What calculations are needed to arrive at it
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Good
Can someone give me problems that involes radical expressions like area,volume or motion of pendulum with solution