<< Chapter < Page Chapter >> Page >

What we have here is, in fact, another conservation law. If the net torque is zero , then angular momentum is constant or conserved . We can see this rigorously by considering net τ = Δ L Δ t size 12{"net "τ= { {ΔL} over {Δt} } } {} for the situation in which the net torque is zero. In that case,

net τ = 0 size 12{"net "τ=0} {}

implying that

Δ L Δ t = 0 . size 12{ { {ΔL} over {Δt} } =0} {}

If the change in angular momentum Δ L size 12{ΔL} {} is zero, then the angular momentum is constant; thus,

L = constant net τ = 0 size 12{L="constant " left ("net "τ=0 right )} {}

or

L = L net τ = 0 . size 12{L=L'" " left ("net "τ=0 right )} {}

These expressions are the law of conservation of angular momentum    . Conservation laws are as scarce as they are important.

An example of conservation of angular momentum is seen in [link] , in which an ice skater is executing a spin. The net torque on her is very close to zero, because there is relatively little friction between her skates and the ice and because the friction is exerted very close to the pivot point. (Both F size 12{F} {} and r size 12{r} {} are small, and so τ size 12{τ} {} is negligibly small.) Consequently, she can spin for quite some time. She can do something else, too. She can increase her rate of spin by pulling her arms and legs in. Why does pulling her arms and legs in increase her rate of spin? The answer is that her angular momentum is constant, so that

L = L . size 12{L=L'} {}

Expressing this equation in terms of the moment of inertia,

= I ω , size 12{Iω=I'ω'} {}

where the primed quantities refer to conditions after she has pulled in her arms and reduced her moment of inertia. Because I size 12{I'} {} is smaller, the angular velocity ω size 12{ω'} {} must increase to keep the angular momentum constant. The change can be dramatic, as the following example shows.

The image a shows an ice skater spinning on the tip of her skate with both her arms and one leg extended. The image b shows the ice skater spinning on the tip of one skate, with her arms crossed and one leg supported on another.
(a) An ice skater is spinning on the tip of her skate with her arms extended. Her angular momentum is conserved because the net torque on her is negligibly small. In the next image, her rate of spin increases greatly when she pulls in her arms, decreasing her moment of inertia. The work she does to pull in her arms results in an increase in rotational kinetic energy.

Calculating the angular momentum of a spinning skater

Suppose an ice skater, such as the one in [link] , is spinning at 0.800 rev/ s with her arms extended. She has a moment of inertia of 2 . 34 kg m 2 size 12{2 "." "34"`"kg" cdot m rSup { size 8{2} } } {} with her arms extended and of 0 . 363 kg m 2 size 12{0 "." "363"`"kg" cdot m rSup { size 8{2} } } {} with her arms close to her body. (These moments of inertia are based on reasonable assumptions about a 60.0-kg skater.) (a) What is her angular velocity in revolutions per second after she pulls in her arms? (b) What is her rotational kinetic energy before and after she does this?

Strategy

In the first part of the problem, we are looking for the skater’s angular velocity ω size 12{ { {ω}} sup { ' }} {} after she has pulled in her arms. To find this quantity, we use the conservation of angular momentum and note that the moments of inertia and initial angular velocity are given. To find the initial and final kinetic energies, we use the definition of rotational kinetic energy given by

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

Solution for (a)

Because torque is negligible (as discussed above), the conservation of angular momentum given in = I ω size 12{Iω= { {I}} sup { ' } { {ω}} sup { ' }} {} is applicable. Thus,

L = L size 12{L=L'} {}

or

= I ω size 12{Iω=I'ω'} {}

Solving for ω and substituting known values into the resulting equation gives

ω = I I ω = 2.34 kg m 2 0 .363 kg m 2 0.800 rev/s = 5.16 rev/s.

Questions & Answers

2 how heat loss is prevented in a vacuum flask
Abdullah Reply
what is science
Helen
logical reasoning for a particular phenomenon.
Ajay
I don't know anything about it 😔. I'm sorry, please forgive 😔
Adarsh
due to non in contact mean no conduction and no convection bec of non conducting base and walls and also their is a grape between the layer like to take the example of thermo flask
Abdul
dimensions v²=u²+2at
Lagben Reply
what if time is not given in finding the average velocity?
Alan Reply
the magnetic circuit of a certain of the flux paths in each of the long and short sides being 25cm and 20cm reprectielectrove. there is an air gap of 2mm long in one the long sides if a flux density of 0.8weber/m is to produce in the magnet of 1500 turns..
Daniel Reply
How do you calculate precision
Sacky Reply
what module is that?
Fillemon
Chemisty 1A?
Fillemon
No it has something to do with measurements bro... What we did today in class
Sacky
Tah bra honestly I didn't understand a thing in that class..when re your Tutorials?
Fillemon
Friday bro... But the topics we did are in this app... Just try to master them quickly before the test dates... Are you done with the Maths sheet
Sacky
I eat ass
Anderson
I'll work on the maths sheet tomorrow bra @Sacky Malyenge but I'll try mastering them
Fillemon
I'll eat your mom's ass with a side of tendies
Anderson
@Fillemon Nanwaapo
Anderson
lol, hush
Emi
There are very large numbers of charged particles in most objects. Why, then, don’t most objects exhibit static electricity?
Bilkisu Reply
Because there's an equal number of negative and positive charges... objects are neutral in nature
NELSON
when a ball rolls on a smooth level ground,the motion of its centre is?
Mary Reply
what is electro magnetic field?
Mary
electromagnetic field is a special type of field been produced by electric charges..!!! like the word electro from Electricity and the word magnetic from Magnetism.. so it is more of a join field..!!!
NELSON
Electromagnetic field is caused by moving electric charge
Muhammad
when a ball rolls on a smooth level ground,the motion of its centre is?
Mumeh
what's the relationship btw displacement and position
Declan Reply
displacement is the change of position 8======✊=D 💦💦
Anderson
what is the meaning of elasticity
Pele Reply
is the ability of a material to or any object to expand to a limit point
king
this is about kinematics you bonk
Emi
what does emf/R mean
Eze Reply
What is work
Wisdom Reply
work is the product of force and perpendicular distance
DAVID
Pls explain simple harmonic motion
Olaiya Reply
Any to and from motion of a fluid or any elastic object
Sacky
a current of 5.5mA is flowing through a 3.3k resistor.compute th p.d developed across the resistor
Clifford Reply
A p.d of 24 volts exist across a 15 OHM'S resistor.calculate the current flowing the resistor
Clifford
a current of 5.5mA is flowing through a 3.3kOHM'S resistor.compute th p.d developed across the resistor
Clifford
solve it please
Festus
the so unit power is the watt(w)/joul/second (w1)/s
Jibo Reply
Really
Lawal
what is time
Jibo Reply
a measure of the duration of an event
Raymond
density
Masente
Practice Key Terms 2

Get the best College physics course in your pocket!





Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask