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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

where

  • 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

where

  • 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?

Solution:

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

I only see partial conversation and what's the question here!
Crow Reply
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RAW Reply
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Damian
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Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
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LITNING
scanning tunneling microscope
Sahil
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
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Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
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Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian 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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