# 0.1 8.2 - revisit angular acceleration  (Page 2/5)

 Page 2 / 5

If the bicycle in the preceding example had been on its wheels instead of upside-down, it would first have accelerated along the ground and then come to a stop. This connection between circular motion and linear motion needs to be explored. For example, it would be useful to know how linear and angular acceleration are related. In circular motion, linear acceleration is tangent to the circle at the point of interest, as seen in [link] . Thus, linear acceleration is called tangential acceleration     ${a}_{\text{t}}$ .

Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction. We know from Uniform Circular Motion and Gravitation that in circular motion centripetal acceleration, ${a}_{\text{c}}$ , refers to changes in the direction of the velocity but not its magnitude. An object undergoing circular motion experiences centripetal acceleration, as seen in [link] . Thus, ${a}_{\text{t}}$ and ${a}_{\text{c}}$ are perpendicular and independent of one another. Tangential acceleration ${a}_{\text{t}}$ is directly related to the angular acceleration $\alpha$ and is linked to an increase or decrease in the velocity, but not its direction.

Now we can find the exact relationship between linear acceleration ${a}_{\text{t}}$ and angular acceleration $\alpha$ . Because linear acceleration is proportional to a change in the magnitude of the velocity, it is defined (as it was in One-Dimensional Kinematics ) to be

${a}_{\text{t}}=\frac{\Delta v}{\Delta t}\text{.}$

For circular motion, note that $v=\mathrm{r\omega }$ , so that

${a}_{\text{t}}=\frac{\Delta \left(\mathrm{r\omega }\right)}{\Delta t}\text{.}$

The radius $r$ is constant for circular motion, and so $\text{Δ}\left(\mathrm{r\omega }\right)=r\left(\Delta \omega \right)$ . Thus,

${a}_{\text{t}}=r\frac{\Delta \omega }{\Delta t}\text{.}$

By definition, $\alpha =\frac{\Delta \omega }{\Delta t}$ . Thus,

${a}_{\text{t}}=\mathrm{r\alpha },$

or

$\alpha =\frac{{a}_{\text{t}}}{r}.$

These equations mean that linear acceleration and angular acceleration are directly proportional. The greater the angular acceleration is, the larger the linear (tangential) acceleration is, and vice versa. For example, the greater the angular acceleration of a car’s drive wheels, the greater the acceleration of the car. The radius also matters. For example, the smaller a wheel, the smaller its linear acceleration for a given angular acceleration $\alpha$ .

## Calculating the angular acceleration of a motorcycle wheel

A powerful motorcycle can accelerate from 0 to 30.0 m/s (about 108 km/h) in 4.20 s. What is the angular acceleration of its 0.320-m-radius wheels? (See [link] .)

Strategy

We are given information about the linear velocities of the motorcycle. Thus, we can find its linear acceleration ${a}_{\text{t}}$ . Then, the expression $\alpha =\frac{{a}_{\text{t}}}{r}$ can be used to find the angular acceleration.

Solution

The linear acceleration is

$\begin{array}{lll}{a}_{\text{t}}& =& \frac{\Delta v}{\Delta t}\\ & =& \frac{\text{30.0 m/s}}{\text{4.20 s}}\\ & =& \text{7.14}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}.\end{array}$

We also know the radius of the wheels. Entering the values for ${a}_{\text{t}}$ and $r$ into $\alpha =\frac{{a}_{\text{t}}}{r}$ , we get

$\begin{array}{lll}\alpha & =& \frac{{a}_{\text{t}}}{r}\\ & =& \frac{\text{7.14}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}}{\text{0.320 m}}\\ & =& \text{22.3}\phantom{\rule{0.25em}{0ex}}{\text{rad/s}}^{2}.\end{array}$

Discussion

Units of radians are dimensionless and appear in any relationship between angular and linear quantities.

So far, we have defined three rotational quantities— , and $\alpha$ . These quantities are analogous to the translational quantities , and $a$ . [link] displays rotational quantities, the analogous translational quantities, and the relationships between them.

Rotational and translational quantities
Rotational Translational Relationship
$\theta$ $x$ $\theta =\frac{x}{r}$
$\omega$ $v$ $\omega =\frac{v}{r}$
$\alpha$ $a$ $\alpha =\frac{{a}_{t}}{r}$

Angular acceleration is a vector, having both magnitude and direction. How do we denote its magnitude and direction? Illustrate with an example.

The magnitude of angular acceleration is $\alpha$ and its most common units are ${\text{rad/s}}^{2}$ . The direction of angular acceleration along a fixed axis is denoted by a + or a – sign, just as the direction of linear acceleration in one dimension is denoted by a + or a – sign. For example, consider a gymnast doing a forward flip. Her angular momentum would be parallel to the mat and to her left. The magnitude of her angular acceleration would be proportional to her angular velocity (spin rate) and her moment of inertia about her spin axis.

## Section summary

• Uniform circular motion is the motion with a constant angular velocity $\omega =\frac{\Delta \theta }{\Delta t}$ .
• In non-uniform circular motion, the velocity changes with time and the rate of change of angular velocity (i.e. angular acceleration) is $\alpha =\frac{\Delta \omega }{\Delta t}$ .
• Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction, given as ${a}_{\text{t}}=\frac{\Delta v}{\Delta t}$ .
• For circular motion, note that $v=\mathrm{r\omega }$ , so that
${a}_{\mathrm{\text{t}}}=\frac{\text{Δ}\left(\mathrm{r\omega }\right)}{\Delta t}.$
• The radius r is constant for circular motion, and so $\mathrm{\text{Δ}}\left(\mathrm{r\omega }\right)=r\Delta \omega$ . Thus,
${a}_{\text{t}}=r\frac{\Delta \omega }{\Delta t}.$
• By definition, $\Delta \omega /\Delta t=\alpha$ . Thus,
${a}_{\text{t}}=\mathrm{r\alpha }$

or

$\alpha =\frac{{a}_{\text{t}}}{r}.$

## Conceptual questions

Analogies exist between rotational and translational physical quantities. Identify the rotational term analogous to each of the following: acceleration, force, mass, work, translational kinetic energy, linear momentum, impulse.

Suppose a piece of food is on the edge of a rotating microwave oven plate. Does it experience nonzero tangential acceleration, centripetal acceleration, or both when: (a) The plate starts to spin? (b) The plate rotates at constant angular velocity? (c) The plate slows to a halt?

## Problems&Exercises

At its peak, a tornado is 60.0 m in diameter and carries 500 km/h winds. What is its angular velocity in revolutions per second?

$\omega =0\text{.}\text{737 rev/s}$

Integrated Concepts

An ultracentrifuge accelerates from rest to 100,000 rpm in 2.00 min. (a) What is its angular acceleration in ${\text{rad/s}}^{2}$ ? (b) What is the tangential acceleration of a point 9.50 cm from the axis of rotation? (c) What is the radial acceleration in ${\text{m/s}}^{2}$ and multiples of $g$ of this point at full rpm?

You have a grindstone (a disk) that is 90.0 kg, has a 0.340-m radius, and is turning at 90.0 rpm, and you press a steel axe against it with a radial force of 20.0 N. (a) Assuming the kinetic coefficient of friction between steel and stone is 0.20, calculate the angular acceleration of the grindstone. (b) How many turns will the stone make before coming to rest?

(a) $-0\text{.}{\text{26 rad/s}}^{2}$

(b) $\text{27}\phantom{\rule{0.25em}{0ex}}\text{rev}$

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
Do somebody tell me a best nano engineering book for beginners?
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
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!