<< Chapter < Page Chapter >> Page >
  • Understand the analogy between angular momentum and linear momentum.
  • Observe the relationship between torque and angular momentum.
  • Apply the law of conservation of angular momentum.

Why does Earth keep on spinning? What started it spinning to begin with? And how does an ice skater manage to spin faster and faster simply by pulling her arms in? Why does she not have to exert a torque to spin faster? Questions like these have answers based in angular momentum, the rotational analog to linear momentum.

By now the pattern is clear—every rotational phenomenon has a direct translational analog. It seems quite reasonable, then, to define angular momentum     L size 12{L} {} as

L = . size 12{L=Iω} {}

This equation is an analog to the definition of linear momentum as p = mv size 12{p= ital "mv"} {} . Units for linear momentum are kg m /s size 12{"kg" cdot m rSup { size 8{2} } "/s"} {} while units for angular momentum are kg m 2 /s size 12{"kg" cdot m rSup { size 8{2} } "/s"} {} . As we would expect, an object that has a large moment of inertia I size 12{I} {} , such as Earth, has a very large angular momentum. An object that has a large angular velocity ω size 12{ω} {} , such as a centrifuge, also has a rather large angular momentum.

Making connections

Angular momentum is completely analogous to linear momentum, first presented in Uniform Circular Motion and Gravitation . It has the same implications in terms of carrying rotation forward, and it is conserved when the net external torque is zero. Angular momentum, like linear momentum, is also a property of the atoms and subatomic particles.

Calculating angular momentum of the earth

Strategy

No information is given in the statement of the problem; so we must look up pertinent data before we can calculate L = size 12{L=Iω} {} . First, according to [link] , the formula for the moment of inertia of a sphere is

I = 2 MR 2 5 size 12{I= { {2 ital "MR" rSup { size 8{2} } } over {5} } } {}

so that

L = = 2 MR 2 ω 5 . size 12{L=Iω= { {2 ital "MR" rSup { size 8{2} } ω} over {5} } } {}

Earth’s mass M size 12{M} {} is 5 . 979 × 10 24 kg size 12{5 "." "979" times "10" rSup { size 8{"24"} } "kg"} {} and its radius R size 12{R} {} is 6 . 376 × 10 6 m size 12{6 "." "376" times "10" rSup { size 8{6} } m} {} . The Earth’s angular velocity ω size 12{ω} {} is, of course, exactly one revolution per day, but we must covert ω size 12{ω} {} to radians per second to do the calculation in SI units.

Solution

Substituting known information into the expression for L size 12{L} {} and converting ω size 12{ω} {} to radians per second gives

L = 0 . 4 5 . 979 × 10 24 kg 6 . 376 × 10 6 m 2 1 rev d = 9 . 72 × 10 37 kg m 2 rev/d . alignl { stack { size 12{L=0 "." 4 left (5 "." "979" times "10" rSup { size 8{"24"} } " kg" right ) left (6 "." "376" times "10" rSup { size 8{6} } " m" right ) rSup { size 8{2} } left ( { {1" rev"} over {d} } right )} {} #" "=9 "." "72" times "10" rSup { size 8{"37"} } " kg" cdot m rSup { size 8{2} } "rev/d" {} } } {}

Substituting size 12{2π} {} rad for 1 size 12{1} {} rev and 8 . 64 × 10 4 s size 12{8 "." "64" times "10" rSup { size 8{4} } s} {} for 1 day gives

L = 9 . 72 × 10 37 kg m 2 rad/rev 8 . 64 × 10 4 s/d 1 rev/d = 7 . 07 × 10 33 kg m 2 /s . alignl { stack { size 12{L= left (9 "." "72" times "10" rSup { size 8{"37"} } " kg" cdot m rSup { size 8{2} } right ) left ( { {2π" rad/rev"} over {8 "." "64" times "10" rSup { size 8{4} } " s/d"} } right ) left (1" rev/d" right )} {} #" "=7 "." "07" times "10" rSup { size 8{"33"} } " kg" cdot m rSup { size 8{2} } "/s" {} } } {}

Discussion

This number is large, demonstrating that Earth, as expected, has a tremendous angular momentum. The answer is approximate, because we have assumed a constant density for Earth in order to estimate its moment of inertia.

When you push a merry-go-round, spin a bike wheel, or open a door, you exert a torque. If the torque you exert is greater than opposing torques, then the rotation accelerates, and angular momentum increases. The greater the net torque, the more rapid the increase in L size 12{L} {} . The relationship between torque and angular momentum is

net τ = Δ L Δ t . size 12{"net "τ= { {ΔL} over {Δt} } } {}

This expression is exactly analogous to the relationship between force and linear momentum, F = Δ p / Δ t size 12{F=Δp/Δt} {} . The equation net τ = Δ L Δ t size 12{"net "τ= { {ΔL} over {Δt} } } {} is very fundamental and broadly applicable. It is, in fact, the rotational form of Newton’s second law.

Calculating the torque putting angular momentum into a lazy susan

[link] shows a Lazy Susan food tray being rotated by a person in quest of sustenance. Suppose the person exerts a 2.50 N force perpendicular to the lazy Susan’s 0.260-m radius for 0.150 s. (a) What is the final angular momentum of the lazy Susan if it starts from rest, assuming friction is negligible? (b) What is the final angular velocity of the lazy Susan, given that its mass is 4.00 kg and assuming its moment of inertia is that of a disk?

The given figure shows a lazy Susan on which various eatables like cake, salad grapes, and a drink are kept. A hand is shown that applies a force F, indicated by a leftward pointing horizontal arrow. This force is perpendicular to the radius r and thus tangential to the circular lazy Susan.
A partygoer exerts a torque on a lazy Susan to make it rotate. The equation net τ = Δ L Δ t size 12{"net "τ= { {ΔL} over {Δt} } } {} gives the relationship between torque and the angular momentum produced.

Strategy

We can find the angular momentum by solving net τ = Δ L Δ t size 12{"net "τ= { {ΔL} over {Δt} } } {} for Δ L size 12{ΔL} {} , and using the given information to calculate the torque. The final angular momentum equals the change in angular momentum, because the lazy Susan starts from rest. That is, Δ L = L size 12{ΔL=L} {} . To find the final velocity, we must calculate ω size 12{ω} {} from the definition of L size 12{L} {} in L = size 12{L=Iω} {} .

Solution for (a)

Solving net τ = Δ L Δ t size 12{"net "τ= { {ΔL} over {Δt} } } {} for Δ L size 12{ΔL} {} gives

Δ L = net τ Δt . size 12{ΔL= left ("net "τ right ) cdot Δt} {}

Because the force is perpendicular to r size 12{r} {} , we see that net τ = rF size 12{"net "τ= ital "rF"} {} , so that

L = rF Δ t = ( 0 . 260 m ) ( 2.50 N ) ( 0.150 s ) = 9 . 75 × 10 2 kg m 2 / s .

Solution for (b)

The final angular velocity can be calculated from the definition of angular momentum,

L = . size 12{L=Iω} {}

Solving for ω size 12{ω} {} and substituting the formula for the moment of inertia of a disk into the resulting equation gives

ω = L I = L 1 2 MR 2 . size 12{ω= { {L} over {I} } = { {L} over { { size 8{1} } wideslash { size 8{2} } ital "MR" rSup { size 8{2} } } } } {}

And substituting known values into the preceding equation yields

ω = 9 . 75 × 10 2 kg m 2 /s 0 . 500 4 . 00 kg 0 . 260 m = 0 . 721 rad/s . size 12{ω= { {9 "." "75" times "10" rSup { size 8{ - 2} } " kg" cdot m rSup { size 8{2} } "/s"} over { left (0 "." "500" right ) left (4 "." "00"" kg" right ) left (0 "." "260"" m" right )} } =0 "." "721"" rad/s"} {}

Discussion

Note that the imparted angular momentum does not depend on any property of the object but only on torque and time. The final angular velocity is equivalent to one revolution in 8.71 s (determination of the time period is left as an exercise for the reader), which is about right for a lazy Susan.

Questions & Answers

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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Introduction to applied math and physics. OpenStax CNX. Oct 04, 2012 Download for free at http://cnx.org/content/col11426/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Introduction to applied math and physics' conversation and receive update notifications?

Ask