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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

why static friction is greater than Kinetic friction
Ali Reply
draw magnetic field pattern for two wire carrying current in the same direction
Ven Reply
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nkombo Reply
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nkombo
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Abid Reply
The potential difference between any two points on the surface is zero that implies È.Ŕ=0, Where R is the distance between two different points &E= Electric field intensity. From which we have cos þ =0, where þ is the angle between the directions of field and distance line, as E andR are zero. Thus
MAHADEV
sorry..E and R are non zero...
MAHADEV
By how much leeway (both percentage and mass) would you have in the selection of the mass of the object in the previous problem if you did not wish the new period to be greater than 2.01 s or less than 1.99 s?
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Myanmar
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what is science
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it's a natural phenomena
Hassan
sap
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please can someone help me with explanations of wave
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there are seven basic type of wave radio waves, gyamma rays (nuclear energy), microwave,etc you can also search 🔍 on Google :-)
Shravasti
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Musa Reply
what is physics
Caya Reply
it is the science which we used in our daily life
Sujitha
Physics is the branch of science that deals with the study of matter and the interactions it undergoes with energy
Junior
it is branch of science which deals with study of happening in the human life
AMIT
A 20MH coil has a resistance of 50 ohms and is connected in series with a capacitor to a 250MV supply if the circuit is to resonate at 100KHZ, Determine 1: the capacitance of the capacitor 2: the working voltage of the circuit, given that pie =3.142
Musa
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Astronomy
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what is physic3
Kalilu
what is physic
Kalilu
Physics? Is a branch of science dealing with matter in relation to energy.
Moses
Physic... Is a purging medicine, which stimulates evacuation of the bowels.
Moses
are you asking for qualities or quantities?
Noman
fundamental quantities are, length , mass, time, current, luminous intensity, amount of substance, thermodynamic temperature.
Shravasti
fundamental quantities are quantities that are independent of others and cannot be define in terms of other quantities there is nothing like Qualities we have only fundamental quantities which includes; length,mass,time, electric current, luminous density, temperature, amount of substance etc
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give examples of three dimensional frame of reference
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Yes the Universe itself
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Lathan Reply
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Noman
thong sleepers are usually used in restrooms.
Noman
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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