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KE rot = 1 2 2 . size 12{"KE" rSub { size 8{"rot"} } = { {1} over {2} } Iω rSup { size 8{2} } } {}

The expression for rotational kinetic energy is exactly analogous to translational kinetic energy, with I size 12{I} {} being analogous to m size 12{m} {} and ω size 12{ω} {} to v size 12{v} {} . Rotational kinetic energy has important effects. Flywheels, for example, can be used to store large amounts of rotational kinetic energy in a vehicle, as seen in [link] .

The figure shows a bus carrying a large flywheel on its board in which rotational kinetic energy is stored.
Experimental vehicles, such as this bus, have been constructed in which rotational kinetic energy is stored in a large flywheel. When the bus goes down a hill, its transmission converts its gravitational potential energy into KE rot size 12{ ital "KE" rSub { size 8{ ital "rot"} } } {} . It can also convert translational kinetic energy, when the bus stops, into KE rot size 12{ ital "KE" rSub { size 8{ ital "rot"} } } {} . The flywheel’s energy can then be used to accelerate, to go up another hill, or to keep the bus from going against friction.

Calculating the work and energy for spinning a grindstone

Consider a person who spins a large grindstone by placing her hand on its edge and exerting a force through part of a revolution as shown in [link] . In this example, we verify that the work done by the torque she exerts equals the change in rotational energy. (a) How much work is done if she exerts a force of 200 N through a rotation of 1.00 rad ( 57.3º ) size 12{1 "." "00"`"rad" \( "57" "." 3 \) rSup { size 8{ circ } } } {} ? The force is kept perpendicular to the grindstone’s 0.320-m radius at the point of application, and the effects of friction are negligible. (b) What is the final angular velocity if the grindstone has a mass of 85.0 kg? (c) What is the final rotational kinetic energy? (It should equal the work.)

Strategy

To find the work, we can use the equation net W = net τ θ size 12{"net "W= left ("net "τ right )θ} {} . We have enough information to calculate the torque and are given the rotation angle. In the second part, we can find the final angular velocity using one of the kinematic relationships. In the last part, we can calculate the rotational kinetic energy from its expression in KE rot = 1 2 2 size 12{"KE" rSub { size 8{"rot"} } = { {1} over {2} } Iω rSup { size 8{2} } } {} .

Solution for (a)

The net work is expressed in the equation

net W = net τ θ , size 12{"net "W= left ("net "τ right )θ} {}

where net τ size 12{τ} {} is the applied force multiplied by the radius ( rF ) size 12{ \( ital "rF" \) } {} because there is no retarding friction, and the force is perpendicular to r size 12{r} {} . The angle θ size 12{θ} {} is given. Substituting the given values in the equation above yields

net W = rF θ = 0.320 m 200 N 1.00 rad = 64.0 N m.

Noting that 1 N · m = 1 J ,

net W = 64.0 J . size 12{"net "W="64" "." 0" J"} {}
The figure shows a large grindstone of radius r which is being given a spin by applying a force F in a counterclockwise direction, as indicated by the arrows.
A large grindstone is given a spin by a person grasping its outer edge.

Solution for (b)

To find ω size 12{ω} {} from the given information requires more than one step. We start with the kinematic relationship in the equation

ω 2 = ω 0 2 + 2 αθ . size 12{ω rSup { size 8{2} } =ω rSub { size 8{0} rSup { size 8{2} } } +2 ital "αθ"} {}

Note that ω 0 = 0 size 12{ω rSub { size 8{0} } =0} {} because we start from rest. Taking the square root of the resulting equation gives

ω = 2 αθ 1 / 2 . size 12{ω= left (2 ital "αθ" right ) rSup { size 8{1/2} } } {}

Now we need to find α size 12{α} {} . One possibility is

α = net τ I , size 12{α= { {"net "τ} over {I} } } {}

where the torque is

net τ = rF = 0.320 m 200 N = 64.0 N m . size 12{"net "τ= ital "rF"= left (0 "." "320"" m" right ) left ("200"" N" right )="64" "." 0" N" cdot m} {}

The formula for the moment of inertia for a disk is found in [link] :

I = 1 2 MR 2 = 0.5 85.0 kg 0.320 m 2 = 4.352 kg m 2 . size 12{I= { {1} over {2} } ital "MR" rSup { size 8{2} } =0 "." 5 left ("85" "." 0" kg" right ) left (0 "." "320"" m" right ) rSup { size 8{2} } =4 "." "352"" kg" cdot m rSup { size 8{2} } } {}

Substituting the values of torque and moment of inertia into the expression for α size 12{α} {} , we obtain

α = 64 . 0 N m 4.352 kg m 2 = 14.7 rad s 2 . size 12{α= { {"64" "." "0 N" cdot m} over {4 "." "352"" kg" cdot m rSup { size 8{2} } } } ="14" "." 7 { {"rad"} over {s rSup { size 8{2} } } } } {}

Now, substitute this value and the given value for θ size 12{θ} {} into the above expression for ω size 12{ω} {} :

ω = 2 αθ 1 / 2 = 2 14.7 rad s 2 1.00 rad 1 / 2 = 5.42 rad s . size 12{ω= left (2 ital "αθ" right ) rSup { size 8{1/2} } = left [2 left ("14" "." 7 { {"rad"} over {s rSup { size 8{2} } } } right ) left (1 "." "00"" rad" right ) right ] rSup { size 8{1/2} } =5 "." "42" { {"rad"} over {s} } } {}

Solution for (c)

The final rotational kinetic energy is

Questions & Answers

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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
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
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.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
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
?
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
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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