# 8.3 Kinematics of rotational motion

 Page 1 / 4
• Observe the kinematics of rotational motion.
• Derive rotational kinematic equations.
• Evaluate problem solving strategies for rotational kinematics.

Just by using our intuition, we can begin to see how rotational quantities like $\theta$ , $\omega$ , and $\alpha$ are related to one another. For example, if a motorcycle wheel has a large angular acceleration for a fairly long time, it ends up spinning rapidly and rotates through many revolutions. In more technical terms, if the wheel’s angular acceleration $\alpha$ is large for a long period of time $t$ , then the final angular velocity $\omega$ and angle of rotation $\theta$ are large. The wheel’s rotational motion is exactly analogous to the fact that the motorcycle’s large translational acceleration produces a large final velocity, and the distance traveled will also be large.

Kinematics is the description of motion. The kinematics of rotational motion    describes the relationships among rotation angle, angular velocity, angular acceleration, and time. Let us start by finding an equation relating $\omega$ , $\alpha$ , and $t$ . To determine this equation, we recall a familiar kinematic equation for translational, or straight-line, motion:

Note that in rotational motion $a={a}_{\text{t}}$ , and we shall use the symbol $a$ for tangential or linear acceleration from now on. As in linear kinematics, we assume $a$ is constant, which means that angular acceleration $\alpha$ is also a constant, because $a=\mathrm{r\alpha }$ . Now, let us substitute $v=\mathrm{r\omega }$ and $a=\mathrm{r\alpha }$ into the linear equation above:

$\mathrm{r\omega }={\mathrm{r\omega }}_{0}+\mathrm{r\alpha t}.$

The radius $r$ cancels in the equation, yielding

where ${\omega }_{0}$ is the initial angular velocity. This last equation is a kinematic relationship among $\omega$ , $\alpha$ , and $t$ —that is, it describes their relationship without reference to forces or masses that may affect rotation. It is also precisely analogous in form to its translational counterpart.

## Making connections

Kinematics for rotational motion is completely analogous to translational kinematics, first presented in One-Dimensional Kinematics . Kinematics is concerned with the description of motion without regard to force or mass. We will find that translational kinematic quantities, such as displacement, velocity, and acceleration have direct analogs in rotational motion.

Starting with the four kinematic equations we developed in One-Dimensional Kinematics , we can derive the following four rotational kinematic equations (presented together with their translational counterparts):

Rotational kinematic equations
Rotational Translational
$\theta =\overline{\omega }t$ $x=\stackrel{-}{v}t$
$\omega ={\omega }_{0}+\mathrm{\alpha t}$ $v={v}_{0}+\text{at}$ (constant $\alpha$ , $a$ )
$\theta ={\omega }_{0}t+\frac{1}{2}{\mathrm{\alpha t}}^{2}$ $x={v}_{0}t+\frac{1}{2}{\text{at}}^{2}$ (constant $\alpha$ , $a$ )
${\omega }^{2}={{\omega }_{0}}^{2}+2\text{αθ}$ ${v}^{2}={{v}_{0}}^{2}+2\text{ax}$ (constant $\alpha$ , $a$ )

In these equations, the subscript 0 denotes initial values ( ${\theta }_{0}$ , ${x}_{0}$ , and ${t}_{0}$ are initial values), and the average angular velocity $\stackrel{-}{\omega }$ and average velocity $\stackrel{-}{v}$ are defined as follows:

The equations given above in [link] can be used to solve any rotational or translational kinematics problem in which $a$ and $\alpha$ are constant.

## Problem-solving strategy for rotational kinematics

1. Examine the situation to determine that rotational kinematics (rotational motion) is involved . Rotation must be involved, but without the need to consider forces or masses that affect the motion.
2. Identify exactly what needs to be determined in the problem (identify the unknowns) . A sketch of the situation is useful.
3. Make a list of what is given or can be inferred from the problem as stated (identify the knowns) .
4. Solve the appropriate equation or equations for the quantity to be determined (the unknown) . It can be useful to think in terms of a translational analog because by now you are familiar with such motion.
5. Substitute the known values along with their units into the appropriate equation, and obtain numerical solutions complete with units . Be sure to use units of radians for angles.

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
Got questions? Join the online conversation and get instant answers!