# 9.4 Rotational kinetic energy: work and energy revisited  (Page 5/9)

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Assuming no losses due to friction, there is only one force doing work—gravity. Therefore the total work done is the change in kinetic energy. As the cans start moving, the potential energy is changing into kinetic energy. Conservation of energy gives

${\text{PE}}_{\text{i}}={\text{KE}}_{\text{f}}.$

More specifically,

${\text{PE}}_{\text{grav}}={\text{KE}}_{\text{trans}}+{\text{KE}}_{\text{rot}}$

or

$\text{mgh}=\frac{1}{2}{\text{mv}}^{2}+\frac{1}{2}{\mathrm{I\omega }}^{2}.$

So, the initial $\text{mgh}$ is divided between translational kinetic energy and rotational kinetic energy; and the greater $I$ is, the less energy goes into translation. If the can slides down without friction, then $\omega =0$ and all the energy goes into translation; thus, the can goes faster.

## Take-home experiment

Locate several cans each containing different types of food. First, predict which can will win the race down an inclined plane and explain why. See if your prediction is correct. You could also do this experiment by collecting several empty cylindrical containers of the same size and filling them with different materials such as wet or dry sand.

## Calculating the speed of a cylinder rolling down an incline

Calculate the final speed of a solid cylinder that rolls down a 2.00-m-high incline. The cylinder starts from rest, has a mass of 0.750 kg, and has a radius of 4.00 cm.

Strategy

We can solve for the final velocity using conservation of energy, but we must first express rotational quantities in terms of translational quantities to end up with $v$ as the only unknown.

Solution

Conservation of energy for this situation is written as described above:

$\text{mgh}=\frac{1}{2}{\mathit{mv}}^{2}+\frac{1}{2}{\mathrm{I\omega }}^{2}.$

Before we can solve for $v$ , we must get an expression for $I$ from [link] . Because $v$ and $\omega$ are related (note here that the cylinder is rolling without slipping), we must also substitute the relationship $\omega =v/R$ into the expression. These substitutions yield

$\text{mgh}=\frac{1}{2}{\mathit{mv}}^{2}+\frac{1}{2}\left(\frac{1}{2}{\mathit{mR}}^{2}\right)\left(\frac{{v}^{2}}{{R}^{2}}\right).$

Interestingly, the cylinder’s radius $R$ and mass $m$ cancel, yielding

$\text{gh}=\frac{1}{2}{v}^{2}+\frac{1}{4}{v}^{2}=\frac{3}{4}{v}^{2}.$

Solving algebraically, the equation for the final velocity $v$ gives

$v={\left(\frac{4\text{gh}}{3}\right)}^{1/2}.$

Substituting known values into the resulting expression yields

$v={\left[\frac{4\left(9.80\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\right)\left(\text{2.00 m}\right)}{3}\right]}^{1/2}=\text{5.11 m/s}.$

Discussion

Because $m$ and $R$ cancel, the result $v={\left(\frac{4}{3}\text{gh}\right)}^{1/2}$ is valid for any solid cylinder, implying that all solid cylinders will roll down an incline at the same rate independent of their masses and sizes. (Rolling cylinders down inclines is what Galileo actually did to show that objects fall at the same rate independent of mass.) Note that if the cylinder slid without friction down the incline without rolling, then the entire gravitational potential energy would go into translational kinetic energy. Thus, $\frac{1}{2}{\text{mv}}^{2}=\text{mgh}$ and $v=\left(2\text{gh}{\right)}^{1/2}$ , which is 22% greater than $\left(4\text{gh}/3{\right)}^{1/2}$ . That is, the cylinder would go faster at the bottom.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?
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
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