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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.)


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 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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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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