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A relationship between rotational and translational kinetic energy can be derived. That relationship is givenby the expression shown in Figure 2 .

Figure 2 . Relationship between rotational and translational kinetic energy.

Krot = Q*Ktr


  • Krot represents rotational kinetic energy
  • Ktr represents translational kinetic energy
  • Q is a constant that depends on how the mass is geometrically distributed in the rolling object

A very interesting equation

The equation shown in Figure 2 is very interesting. It shows that the relationship that determines the distribution of kinetic energy betweentranslational kinetic energy and rotational kinetic depends solely on the constant Q, and is independent of the mass, the radius, etc.

The constant Q depends on how the mass is geometrically distributed in the rolling object. Generally speaking, the more the mass is distributed toward theouter edge of the object, the greater will be the value of Q.

Total kinetic energy

The total kinetic energy is given by

K = Ktr + Krot, or

K = (1/2)*M*(Vcm)^2 + (1/2)*Icm*W^2


  • K represents total kinetic energy
  • M represents the mass
  • Vcm represents the translational velocity of the center of mass
  • Icm represents the moment of inertia through the center of mass
  • W represents the angular velocity

Through substitution we can write

K = Ktr + Krot, or

K = Ktr + Q*Ktr, or

K = (1 + Q)*Ktr, or

K = (1 + Q)*(1/2)*M*(Vcm)^2

Thus, an object with a given mass and a given translational velocity has a total kinetic energy that is proportional to (1+Q). The larger the value of Q,the greater will be the kinetic energy possessed by the object.

From Figure 1 , a rolling hollow cylinder with a Q value of 1 would have a greater total kinetic energy than a solid cylinder with the same mass and a Qvalue of 0.5 rolling with the same translational velocity. This is because the cylinder would have more rotational kinetic energy.

Acceleration of a rolling object

If there were no friction, an object would not roll down an incline. Instead it would simply slide down the incline. In the absence of torque, the rotationalinertia of the object would prevent it from experiencing angular acceleration.

The frictional force parallel to the incline and pointing up the incline produces a torque that causes the object to experience angular acceleration.

Example scenarios

In this section, I will explain some example scenarios involving various aspects of rolling solid and hollow cylinders.

Rolling cylinders

A cylindrical shell and a solid cylinder, each of unknown mass and unknown radii roll down an incline of unknown height and unknown angle.

Part 1

Which object will have the greater translational velocity when they reach the bottom of the incline?


From Figure 1 , the rotational inertia for a thin hollow cylinder is given by

I = M*R^2

The rotational inertia for a solid cylinder is given by

I = (1/2)*MR^2

Let the cylindrical shell be identified as M1 and the solid cylinder be identified as M2.

The total kinetic for each object at the bottom is equal to potential energy for that object at the top. Therefore,

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Accessible physics concepts for blind students. OpenStax CNX. Oct 02, 2015 Download for free at https://legacy.cnx.org/content/col11294/1.36
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