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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"} } } {}


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.


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.


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"} {}


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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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.
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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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