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We now return to the helix introduced earlier in this chapter. A vector-valued function that describes a helix can be written in the form

r ( t ) = R cos ( 2 π N t h ) i + R sin ( 2 π N t h ) j + t k , 0 t h ,

where R represents the radius of the helix, h represents the height (distance between two consecutive turns), and the helix completes N turns. Let’s derive a formula for the arc length of this helix using [link] . First of all,

r ( t ) = 2 π N R h sin ( 2 π N t h ) i + 2 π N R h cos ( 2 π N t h ) j + k .

Therefore,

s = a b r ( t ) d t = 0 h ( 2 π N R h sin ( 2 π N t h ) ) 2 + ( 2 π N R h cos ( 2 π N t h ) ) 2 + 1 2 d t = 0 h 4 π 2 N 2 R 2 h 2 ( sin 2 ( 2 π N t h ) + cos 2 ( 2 π N t h ) ) + 1 d t = 0 h 4 π 2 N 2 R 2 h 2 + 1 d t = [ t 4 π 2 N 2 R 2 h 2 + 1 ] 0 h = h 4 π 2 N 2 R 2 + h 2 h 2 = 4 π 2 N 2 R 2 + h 2 .

This gives a formula for the length of a wire needed to form a helix with N turns that has radius R and height h.

Arc-length parameterization

We now have a formula for the arc length of a curve defined by a vector-valued function. Let’s take this one step further and examine what an arc-length function    is.

If a vector-valued function represents the position of a particle in space as a function of time, then the arc-length function measures how far that particle travels as a function of time. The formula for the arc-length function follows directly from the formula for arc length:

s ( t ) = a t ( f ( u ) ) 2 + ( g ( u ) ) 2 + ( h ( u ) ) 2 d u .

If the curve is in two dimensions, then only two terms appear under the square root inside the integral. The reason for using the independent variable u is to distinguish between time and the variable of integration. Since s ( t ) measures distance traveled as a function of time, s ( t ) measures the speed of the particle at any given time. Since we have a formula for s ( t ) in [link] , we can differentiate both sides of the equation:

s ( t ) = d d t [ a t ( f ( u ) ) 2 + ( g ( u ) ) 2 + ( h ( u ) ) 2 d u ] = d d t [ a t r ( u ) d u ] = r ( t ) .

If we assume that r ( t ) defines a smooth curve, then the arc length is always increasing, so s ( t ) > 0 for t > a . Last, if r ( t ) is a curve on which r ( t ) = 1 for all t , then

s ( t ) = a t r ( u ) d u = a t 1 d u = t a ,

which means that t represents the arc length as long as a = 0.

Arc-length function

Let r ( t ) describe a smooth curve for t a . Then the arc-length function is given by

s ( t ) = a t r ( u ) d u .

Furthermore, d s d t = r ( t ) > 0. If r ( t ) = 1 for all t a , then the parameter t represents the arc length from the starting point at t = a .

A useful application of this theorem is to find an alternative parameterization of a given curve, called an arc-length parameterization    . Recall that any vector-valued function can be reparameterized via a change of variables. For example, if we have a function r ( t ) = 3 cos t , 3 sin t , 0 t 2 π that parameterizes a circle of radius 3, we can change the parameter from t to 4 t , obtaining a new parameterization r ( t ) = 3 cos 4 t , 3 sin 4 t . The new parameterization still defines a circle of radius 3, but now we need only use the values 0 t π / 2 to traverse the circle once.

Suppose that we find the arc-length function s ( t ) and are able to solve this function for t as a function of s. We can then reparameterize the original function r ( t ) by substituting the expression for t back into r ( t ) . The vector-valued function is now written in terms of the parameter s. Since the variable s represents the arc length, we call this an arc-length parameterization of the original function r ( t ) . One advantage of finding the arc-length parameterization is that the distance traveled along the curve starting from s = 0 is now equal to the parameter s. The arc-length parameterization also appears in the context of curvature (which we examine later in this section) and line integrals, which we study in the Introduction to Vector Calculus .

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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
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Akash Reply
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Maciej
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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
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Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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