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The figure shows a flat surface inclined at a height of h from the surface level, with three cans of soup of different densities numbered as one, two, and three rolling along it.
Three cans of soup with identical masses race down an incline. The first can has a low friction coating and does not roll but just slides down the incline. It wins because it converts its entire PE into translational KE. The second and third cans both roll down the incline without slipping. The second can contains thin soup and comes in second because part of its initial PE goes into rotating the can (but not the thin soup). The third can contains thick soup. It comes in third because the soup rotates along with the can, taking even more of the initial PE for rotational KE, leaving less for translational KE.

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

PE i = KE f . size 12{"PE" rSub { size 8{i} } ="KE" rSub { size 8{f} } } {}

More specifically,

PE grav = KE trans + KE rot size 12{"PE" rSub { size 8{"grav"} } ="KE" rSub { size 8{"trans"} } +"KE" rSub { size 8{"rot"} } } {}

or

mgh = 1 2 mv 2 + 1 2 2 . size 12{ ital "mgh"= { {1} over {2} } ital "mv" rSup { size 8{2} } + { {1} over {2} } Iω rSup { size 8{2} } } {}

So, the initial mgh size 12{ ital "mgh"} {} is divided between translational kinetic energy and rotational kinetic energy; and the greater I size 12{I} {} is, the less energy goes into translation. If the can slides down without friction, then ω = 0 size 12{ω=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:

mgh = 1 2 mv 2 + 1 2 2 . size 12{ ital "mgh"= { {1} over {2} } ital "mv" rSup { size 8{2} } + { {1} over {2} } Iω rSup { size 8{2} } } {}

Before we can solve for v size 12{v} {} , we must get an expression for I size 12{I} {} from [link] . Because v size 12{v} {} and ω size 12{ω} {} are related (note here that the cylinder is rolling without slipping), we must also substitute the relationship ω = v / R size 12{ω=v/R} {} into the expression. These substitutions yield

mgh = 1 2 mv 2 + 1 2 1 2 mR 2 v 2 R 2 . size 12{ ital "mgh"= { {1} over {2} } ital "mv" rSup { size 8{2} } + { {1} over {2} } left ( { {1} over {2} } ital "mR" rSup { size 8{2} } right ) left ( { {v rSup { size 8{2} } } over {R rSup { size 8{2} } } } right )} {}

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

gh = 1 2 v 2 + 1 4 v 2 = 3 4 v 2 . size 12{ ital "gh"= { {1} over {2} } v rSup { size 8{2} } + { {1} over {4} } v rSup { size 8{2} } = { {3} over {4} } v rSup { size 8{2} } } {}

Solving algebraically, the equation for the final velocity v size 12{v} {} gives

v = 4 gh 3 1 / 2 . size 12{v= left ( { {4 ital "gh"} over {3} } right ) rSup { size 8{1/2} } } {}

Substituting known values into the resulting expression yields

v = 4 9.80 m/s 2 2.00 m 3 1 / 2 = 5.11 m/s . size 12{v= left [ { {4 left (9 "." "80"" m/s" rSup { size 8{2} } right ) left (2 "." "00"" m" right )} over {3} } right ] rSup { size 8{1/2} } =5 "." "11"" m/s"} {}

Discussion

Because m size 12{m} {} and R size 12{R} {} cancel, the result v = 4 3 gh 1 / 2 size 12{v= left ( { {4} over {3} } ital "gh" right ) rSup { size 8{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, 1 2 mv 2 = mgh size 12{ \( 1/2 \) ital "mv" rSup { size 8{2} } `= ital "mgh"} {} and v = ( 2 gh ) 1 / 2 size 12{v= \( 2 ital "gh" \) rSup { size 8{1/2} } } {} , which is 22% greater than ( 4 gh / 3 ) 1 / 2 size 12{ \( 4 ital "gh"/3 \) rSup { size 8{1/2} } } {} . That is, the cylinder would go faster at the bottom.

Questions & Answers

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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
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, Mechanics. OpenStax CNX. Apr 15, 2013 Download for free at http://legacy.cnx.org/content/col11506/1.2
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