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Torque is the analog of force and moment of inertia is the analog of mass. Force and mass are physical quantities that depend on only one factor. For example, mass is related solely to the numbers of atoms of various types in an object. Are torque and moment of inertia similarly simple?

No. Torque depends on three factors: force magnitude, force direction, and point of application. Moment of inertia depends on both mass and its distribution relative to the axis of rotation. So, while the analogies are precise, these rotational quantities depend on more factors.

Section summary

  • The farther the force is applied from the pivot, the greater is the angular acceleration; angular acceleration is inversely proportional to mass.
  • If we exert a force F size 12{F} {} on a point mass m size 12{m} {} that is at a distance r size 12{r} {} from a pivot point and because the force is perpendicular to r size 12{r} {} , an acceleration a = F/m size 12{F} {} is obtained in the direction of F size 12{F} {} . We can rearrange this equation such that
    F = ma , size 12{F} {","}

    and then look for ways to relate this expression to expressions for rotational quantities. We note that a = rα size 12{F} {} , and we substitute this expression into F=ma size 12{F} {} , yielding

    F=mrα size 12{F} {}
  • Torque is the turning effectiveness of a force. In this case, because F size 12{F} {} is perpendicular to r size 12{r} {} , torque is simply τ = rF size 12{F} {} . If we multiply both sides of the equation above by r size 12{r} {} , we get torque on the left-hand side. That is,
    rF = mr 2 α size 12{ ital "rF"= ital "mr" rSup { size 8{2} } α} {}


    τ = mr 2 α . size 12{τ= ital "mr" rSup { size 8{2} } α "." } {}
  • The moment of inertia I size 12{I} {} of an object is the sum of MR 2 size 12{ ital "MR" rSup { size 8{2} } } {} for all the point masses of which it is composed. That is,
    I = mr 2 . size 12{I= sum ital "mr" rSup { size 8{2} } "." } {}
  • The general relationship among torque, moment of inertia, and angular acceleration is
    τ = size 12{τ=Iα} {}


    α = net τ I size 12{α= { { ital "net"`τ} over {I} } cdot } {}

Conceptual questions

The moment of inertia of a long rod spun around an axis through one end perpendicular to its length is ML 2 /3 size 12{"ML" rSup { size 8{2} } "/3"} {} . Why is this moment of inertia greater than it would be if you spun a point mass M at the location of the center of mass of the rod (at L / 2 size 12{L/2} {} )? (That would be ML 2 /4 size 12{"ML" rSup { size 8{2} } "/4"} {} .)

Why is the moment of inertia of a hoop that has a mass M and a radius R greater than the moment of inertia of a disk that has the same mass and radius? Why is the moment of inertia of a spherical shell that has a mass M and a radius R greater than that of a solid sphere that has the same mass and radius?

Give an example in which a small force exerts a large torque. Give another example in which a large force exerts a small torque.

While reducing the mass of a racing bike, the greatest benefit is realized from reducing the mass of the tires and wheel rims. Why does this allow a racer to achieve greater accelerations than would an identical reduction in the mass of the bicycle’s frame?

The given figure shows a racing bicycle leaning on a door.
The image shows a side view of a racing bicycle. Can you see evidence in the design of the wheels on this racing bicycle that their moment of inertia has been purposely reduced? (credit: Jesús Rodriguez)

A ball slides up a frictionless ramp. It is then rolled without slipping and with the same initial velocity up another frictionless ramp (with the same slope angle). In which case does it reach a greater height, and why?


This problem considers additional aspects of example Calculating the Effect of Mass Distribution on a Merry-Go-Round . (a) How long does it take the father to give the merry-go-round and child an angular velocity of 1.50 rad/s? (b) How many revolutions must he go through to generate this velocity? (c) If he exerts a slowing force of 300 N at a radius of 1.35 m, how long would it take him to stop them?

(a) 0.338 s

(b) 0.0403 rev

(c) 0.313 s

Questions & Answers

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
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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.
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, Introduction to applied math and physics. OpenStax CNX. Oct 04, 2012 Download for free at http://cnx.org/content/col11426/1.3
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