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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

Pls guys am having problem on these topics: latent heat of fusion, specific heat capacity and the sub topics under them.Pls who can help?
hamidat Reply
Thanks George,I appreciate.
hamidat
this will lead you rightly of the formula to use
Abolarin
Most especially it is the calculatory aspects that is giving me issue, but with these new strength that you guys have given me,I will put in my best to understand it again.
hamidat
you can bring up a question and let's see what we can do to it
Abolarin
the distance between two suasive crests of water wave traveling of 3.6ms1 is 0.45m calculate the frequency of the wave
Idris Reply
v=f×lemda where the velocity is given and lends also given so simply u can calculate the frequency
Abdul
You are right my brother, make frequency the subject of formula and equate the values of velocity and lamda into the equation, that all.
hamidat
lExplain what happens to the energy carried by light that it is dimmed by passing it through two crossed polarizing filters.
Christoper Reply
When light is reflected at Brewster's angle from a smooth surface, it is 100% polarizedparallel to the surface. Part of the light will be refracted into the surface.
Ekram
What is specific heat capacity?
hamidat Reply
Specific heat capacity is the amount of heat required to raise the temperature of one (Kg) of a substance through one Kelvin
Paluutar
formula for measuring Joules
Rowshan Reply
I don't understand, do you mean the S.I unit of work and energy?
hamidat
what are the effects of electric current
ADAMS Reply
What limits the Magnification of an optical instrument?
Naeem Reply
Lithography is 2 micron
Venkateshwarlu
what is expression for energy possessed by water ripple
Prabesh Reply
what is hydrolic press
Mark Reply
An hydraulic press is a type of machine that is operated by different pressure of water on pistons.
hamidat
what is dimensional unite of mah
Patrock Reply
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sharon Reply
What is Boyles law
Pascal Reply
it can simple defined as constant temperature
Muhammad
Boyles law states that the volume of a fixed amount of a gas is inversely proportional to the pressure acting on in provided that the temperature is constant.that is V=k(1/p) or V=k/p
FADILAT
what is motion
Mua Reply
getting notifications for a dictionary word, smh
Anderson
what is escape velocity
Shuaibu Reply
the minimum thrust that an object must have in oder yo escape the gravitational pull
Joshua
what is a dimer
Mua
what is a atom
ADAMS
how to calculate tension
Deena Reply
what are the laws of motion
Mua
Practice Key Terms 2

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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