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T = I*An

We also know that

T = R*f


  • T is the torque
  • R is the radius
  • f is the unknown frictional force


T = I*An, or

An = T/I, or

An = R*f/I, or

f = An*I/R


  • T represents the torque
  • I represents the moment of inertia
  • R represents the radius
  • f represents the unknown frictional force

As we discussed earlier, when an object is rolling with a given angular velocity, the translational velocity of the axis of rotation is proportional tothe radius of the object. The greater the radius, the greater will be the translational velocity. Thus, the translational velocity is proportional to theangular velocity with the radius being the proportionality constant. We can write

Vcm = W*R

If the object is not slipping, the rate of change of the translational velocity must also be proportional to the rate of change of the angular velocitythrough the same proportionality constant, which is the radius.

Therefore, we can write

Atr = R*An, or

An = Atr/R

The translational acceleration is also given by the net force divided by the mass. The net force is the component of the weight pointing down the inclineminus the force of friction pointing up the incline. Therefore, we can write

M*g*sin(U) - f = M*Atr

Substitution from above yields

M*g*sin(U) - (An*I/R) = M*Atr

Further substitution from above yields

M*g*sin(U) - ((Atr/R)*I/R) = M*Atr, or

M*g*sin(U) - (Atr*I/(R^2)) = M*Atr

Solving for Atr yields

M*Atr + (Atr*I/(R^2)) = M*g*sin(U), or

Atr * (M + I/(R^2)) = M*g*sin(U), or

Atr = (M*g*sin(U))/(M + I/(R^2)), or

Atr = (g*sin(U))/(1 + I/(M*R^2))

For a solid cylinder,

I = (1/2)*M*R^2

By substitution

Atr = (g*sin(U))/(1 + ((1/2)*M*R^2)/(M*R^2)), or

Atr = (g*sin(U))/(1 + (1/2)), or

Atr = (g*sin(U))/(3/2), or

Therefore, the translational acceleration of the rolling solid cylinder is given by

Atr = (2/3)*(g*sin(U))

If the cylinder were sliding in the total absence of friction, from what you learned in earlier modules, the acceleration would simply be

Atr = g*sin(U)

Therefore, if there is sufficient friction to cause the cylinder to roll without slipping, the translational acceleration will be only (2/3) of thesliding acceleration. Once again, this is the result of a portion of the potential energy being transformed into rotational kinetic energy, resulting inless translational velocity, and less translational acceleration.

Work through the computations

I encourage you to work through the computations that I have presented in this lesson to confirm that you get the same results. Experiment withthe scenarios, making changes, and observing the results of your changes. Make certain that you can explain why your changes behave as they do.


I will publish a module containing consolidated links to resources on my Connexions web page and will update and add to the list as additional modulesin this collection are published.


This section contains a variety of miscellaneous information.

Housekeeping material
  • Module name: Angular Momentum -- The Physics of Rolling Objects
  • File: Phy1350.htm
  • Revised: 10/02/15
  • Keywords:
    • physics
    • accessible
    • accessibility
    • blind
    • graph board
    • protractor
    • screen reader
    • refreshable Braille display
    • JavaScript
    • trigonometry
    • rotational kinetic energy
    • translational kinetic energy

Financial : Although the openstax CNX site makes it possible for you to download a PDF file for the collection that contains thismodule at no charge, and also makes it possible for you to purchase a pre-printed version of the PDF file, you should beaware that some of the HTML elements in this module may not translate well into PDF.

You also need to know that Prof. Baldwin receives no financial compensation from openstax CNX even if you purchase the PDF version of the collection.

In the past, unknown individuals have copied Prof. Baldwin's modules from cnx.org, converted them to Kindle books, and placed them for sale on Amazon.com showing Prof. Baldwin as the author.Prof. Baldwin neither receives compensation for those sales nor does he know who doesreceive compensation. If you purchase such a book, please be aware that it is a copy of a collection that is freelyavailable on openstax CNX and that it was made and published without the prior knowledge of Prof. Baldwin.

Affiliation : Prof. Baldwin is a professor of Computer Information Technology at Austin Community College in Austin, TX.


Questions & Answers

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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