# Rotation angle and angular velocity

 Page 1 / 9
• Define arc length, rotation angle, radius of curvature and angular velocity.
• Calculate the angular velocity of a car wheel spin.

In Kinematics , we studied motion along a straight line and introduced such concepts as displacement, velocity, and acceleration. Two-Dimensional Kinematics dealt with motion in two dimensions. Projectile motion is a special case of two-dimensional kinematics in which the object is projected into the air, while being subject to the gravitational force, and lands a distance away. In this chapter, we consider situations where the object does not land but moves in a curve. We begin the study of uniform circular motion by defining two angular quantities needed to describe rotational motion.

## Rotation angle

When objects rotate about some axis—for example, when the CD (compact disc) in [link] rotates about its center—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit    used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the rotation angle     $\text{Δ}\theta$ to be the ratio of the arc length to the radius of curvature:

$\text{Δ}\theta =\frac{\text{Δ}s}{r}\text{.}$

The arc length     $\phantom{\rule{0.25em}{0ex}}\text{Δ}s$ is the distance traveled along a circular path as shown in [link] Note that $r$ is the radius of curvature    of the circular path.

We know that for one complete revolution, the arc length is the circumference of a circle of radius $r$ . The circumference of a circle is $2\pi r$ . Thus for one complete revolution the rotation angle is

$\text{Δ}\theta =\frac{2\pi r}{r}=2\pi \text{.}$

This result is the basis for defining the units used to measure rotation angles, $\text{Δ}\theta$ to be radians    (rad), defined so that

$2\pi \phantom{\rule{0.25em}{0ex}}\text{rad}=\text{1 revolution.}$

A comparison of some useful angles expressed in both degrees and radians is shown in [link] .

Comparison of angular units
$\text{30º}$ $\frac{\pi }{6}$
$\text{60º}$ $\frac{\pi }{3}$
$\text{90º}$ $\frac{\pi }{2}$
$\text{120º}$ $\frac{2\pi }{3}$
$\text{135º}$ $\frac{3\pi }{4}$
$\text{180º}$ $\pi$

If $\text{Δ}\theta =2\pi$ rad, then the CD has made one complete revolution, and every point on the CD is back at its original position. Because there are $\text{360º}$ in a circle or one revolution, the relationship between radians and degrees is thus

$2\pi \phantom{\rule{0.25em}{0ex}}\text{rad}=\text{360º}$

so that

$1\phantom{\rule{0.25em}{0ex}}\text{rad}=\frac{\text{360º}}{2\pi }\approx \text{57.}3º\text{.}$

## Angular velocity

How fast is an object rotating? We define angular velocity     $\omega$ as the rate of change of an angle. In symbols, this is

$\omega =\frac{\text{Δ}\theta }{\text{Δ}t}\text{,}$

where an angular rotation $\text{Δ}\theta$ takes place in a time $\text{Δ}t$ . The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad/s).

Angular velocity $\omega$ is analogous to linear velocity $v$ . To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length $\text{Δ}s$ in a time $\text{Δ}t$ , and so it has a linear velocity

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
what is biological synthesis of nanoparticles
Berger describes sociologists as concerned with
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 40, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8048878908157349, "perplexity": 1284.5439188457153}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526536.46/warc/CC-MAIN-20190720153215-20190720175215-00406.warc.gz"}