# 10.1 Rotational variables  (Page 2/11)

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where $\theta$ is the angle of rotation ( [link] ). The units of angular velocity are radians per second (rad/s). Angular velocity can also be referred to as the rotation rate in radians per second. In many situations, we are given the rotation rate in revolutions/s or cycles/s. To find the angular velocity, we must multiply revolutions/s by $2\pi$ , since there are $2\pi$ radians in one complete revolution. Since the direction of a positive angle in a circle is counterclockwise, we take counterclockwise rotations as being positive and clockwise rotations as negative.

We can see how angular velocity is related to the tangential speed of the particle by differentiating [link] with respect to time. We rewrite [link] as

$s=r\theta .$

Taking the derivative with respect to time and noting that the radius r is a constant, we have

$\frac{ds}{dt}=\frac{d}{dt}\left(r\theta \right)=\theta \frac{dr}{dt}+r\frac{d\theta }{dt}=r\frac{d\theta }{dt}$

where $\theta \frac{dr}{dt}=0$ . Here $\frac{ds}{dt}$ is just the tangential speed ${v}_{\text{t}}$ of the particle in [link] . Thus, by using [link] , we arrive at

${v}_{\text{t}}=r\omega .$

That is, the tangential speed of the particle is its angular velocity times the radius of the circle. From [link] , we see that the tangential speed of the particle increases with its distance from the axis of rotation for a constant angular velocity. This effect is shown in [link] . Two particles are placed at different radii on a rotating disk with a constant angular velocity. As the disk rotates, the tangential speed increases linearly with the radius from the axis of rotation. In [link] , we see that ${v}_{1}={r}_{1}{\omega }_{1}$ and ${v}_{2}={r}_{2}{\omega }_{2}$ . But the disk has a constant angular velocity, so ${\omega }_{1}={\omega }_{2}$ . This means $\frac{{v}_{1}}{{r}_{1}}=\frac{{v}_{2}}{{r}_{2}}$ or ${v}_{2}=\left(\frac{{r}_{2}}{{r}_{1}}\right){v}_{1}$ . Thus, since ${r}_{2}>{r}_{1}$ , ${v}_{2}>{v}_{1}$ .

Up until now, we have discussed the magnitude of the angular velocity $\omega =d\theta \text{/}dt,$ which is a scalar quantity—the change in angular position with respect to time. The vector $\stackrel{\to }{\omega }$ is the vector associated with the angular velocity and points along the axis of rotation. This is useful because when a rigid body is rotating, we want to know both the axis of rotation and the direction that the body is rotating about the axis, clockwise or counterclockwise. The angular velocity $\stackrel{\to }{\omega }$ gives us this information. The angular velocity $\stackrel{\to }{\omega }$ has a direction determined by what is called the right-hand rule. The right-hand rule is such that if the fingers of your right hand wrap counterclockwise from the x -axis (the direction in which $\theta$ increases) toward the y- axis, your thumb points in the direction of the positive z -axis ( [link] ). An angular velocity $\stackrel{\to }{\omega }$ that points along the positive z -axis therefore corresponds to a counterclockwise rotation, whereas an angular velocity $\stackrel{\to }{\omega }$ that points along the negative z -axis corresponds to a clockwise rotation.

We can verify the right-hand-rule using the vector expression for the arc length $\stackrel{\to }{s}=\stackrel{\to }{\theta }\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{r}$ , [link] . If we differentiate this equation with respect to time, we find

$\frac{d\stackrel{\to }{s}}{dt}=\frac{d}{dt}\left(\stackrel{\to }{\theta }\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{r}\right)=\left(\frac{d\stackrel{\to }{\theta }}{dt}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{r}\right)+\left(\stackrel{\to }{\theta }\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{d\stackrel{\to }{r}}{dt}\right)=\frac{d\stackrel{\to }{\theta }}{dt}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{r}.$

who discover periodic table?
it wasn't discovered , it was made.and the person who made it was dmitri mendeleev. dobreinier and newland gave their laws before dmitri related periodic table but wasn't successful in their work
Ritik
Nope, numerous number of scientist had actually contributed in the making of periodic table. Dmitri Mendeleev succeeded making all the elements into the right order in accordance to their atomic number.
Dame
what is the Greek name for calcium
different types of wave
longitudinal and transverse waves
Ravindra
a gun is kept in the state that it cannot move anywhere and the bullet is fired. Then what is the effect on the velocity of bullet and KE of gun ?
want is meant by the term solar system
it refers to the sun and all heavenly bodies revolving around it.
Danie
excatly...for sure
Arzoodan
in addition to Danie's, a solar system is a collection of planets and their moons, asteroids, and other objects bound together by the Star's gravitational force directly or indirectly.
Galiwango
what is meant by total internal reflection
what iw meant by total internal reflection
Akshay
Lorentz force?
jyotirmayee
study fibre optics. .you will get total internal reflection
Siddhansh
wha
jyotirmayee
what is Lorentz force?
jyotirmayee
a ray of light traveling at an angle of incidence greater than critical angle from denser to rarer medium is totally reflected back into the denser medium is called total internal reflaction
Manoj
motion in strat line where is this chapter
motion in straight line is kinematic's part
Ritik
yea
Manoj
this defination isn't correct
Arzoodan
motion in one dimension
Anil
what is Lorentz force?
what is maxwell electromagnetic law?
jyotirmayee
at what angle should the two forces 2p and root 2p acts so that the resultant force is p root 10
Akshay
what's the working difference between a dynamo and a pump?
a dynamo is basically a dc generator while pump is usually equipped with a motor
vedanth
a dynamo converts mechanical energy to electrical while a pump is opposite to that
vedanth
nice
Piyali
okay
Friday
why sea water looks bluish?
cuz the sky is blue...
Mehdi
you see the reflexion of the "blue" sky in the water
Mehdi
somewhere sea water turns green why?
Piyali
never seen bro... are u sure ?
Mehdi
ur answer was correct but due to the presence of phytoplankton color can be changed near the shore
Piyali
waaaww... you re awesome
Mehdi
because of the reflection of the sky
Friday
rays coming from the sun consist of all 7 colours ie.VIBGYOR. when the ray strikes surface of water,all colors gets absorbed by it except blue which gets reflected by it.so we find the sea water appearing bluish
Ritik
how can someone identify sea water from rain water
Oniyide
i think it's not possible as because rainwater consists of water from all kind of water bodies ie.lakes,seas etc but u can predict if u have a sea nearby ur home or city
Ritik
i need solutions of unuversity pbysucs volume 1
me too
Nirupam
Help us if anyone knows
Nirupam
bring questions
john
actually if u wanted whole book solution then u should buy the solution book
jyotirmayee
from where
Nirupam
where do u live ,,,,,if u live in Delhi then at bellsarayeiii or from stationary store ,,,,chatarpurrr ,,,,,there is a popular books store,,,,,u have to buy from there
jyotirmayee
Kota Rajasthan
Nirupam
so strange,,,,,r u preparing for pmt,,,?
jyotirmayee
for IIT
Nirupam
then concern near book store
jyotirmayee
in Rajasthan
jyotirmayee
jyotirmayee
Nirupam
its fine,,most welcome
jyotirmayee
what is three dimensional coordinate system?
considering the change of vectors in all three dimensions of space
vedanth
direction co-sign of vector questions
opposite
Sonu
opposite mean
Vipin
what is the different between action and reaction?
action is external force. Reaction exists because of the external action. Reaction is mostly an internal force
Yoblaze
The difference is said in the word itself. There is no existance of reaction without an application of action. So action occur first then reaction. Reaction may produce in body itself or to other body i.e., it may be internal or external.
Amalesh
or we can say , reaction is the result of action
Ritik