# 10.5 Angular momentum and its conservation  (Page 3/7)

 Page 3 / 7

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 $\text{net}\phantom{\rule{0.25em}{0ex}}\tau =\frac{\text{Δ}L}{\text{Δ}t}$ for the situation in which the net torque is zero. In that case,

$\text{net}\tau =0$

implying that

$\frac{\text{Δ}L}{\text{Δ}t}=0.$

If the change in angular momentum $\text{Δ}L$ is zero, then the angular momentum is constant; thus,

$L=\text{constant}\phantom{\rule{0.25em}{0ex}}\left(\text{net}\phantom{\rule{0.25em}{0ex}}\tau =0\right)$

or

$L=L\prime \text{}\left(\text{net}\tau =0\right).$

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 $F$ and $r$ are small, and so $\tau$ 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

$L=L\prime .$

Expressing this equation in terms of the moment of inertia,

$\mathrm{I\omega }=I\prime \omega \prime ,$

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

## 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 $2\text{.}\text{34}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot {\text{m}}^{2}$ with her arms extended and of $0\text{.}\text{363}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot {\text{m}}^{2}$ 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 $\omega \prime$ 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

${\text{KE}}_{\text{rot}}=\frac{1}{2}{\mathrm{I\omega }}^{2}.$

Solution for (a)

Because torque is negligible (as discussed above), the conservation of angular momentum given in $\mathrm{I\omega }=I\prime \omega \prime$ is applicable. Thus,

$L=L\prime$

or

$\mathrm{I\omega }=I\prime \omega \prime$

Solving for $\omega \prime$ and substituting known values into the resulting equation gives

$\begin{array}{lll}\omega \prime & =& \frac{I}{I\prime }\omega =\left(\frac{\text{2.34 kg}\cdot {m}^{2}}{0\text{.363 kg}\cdot {m}^{2}}\right)\left(\text{0.800 rev/s}\right)\\ & =& \text{}\text{5.16 rev/s.}\end{array}$

how many subject is in physics
the write question should be " How many Topics are in O- Level Physics, or other branches of physics.
effiom
how many topic are in physics
Praise
yh I need someone to explain something im tryna solve . I'll send the question if u down for it
a ripple tank experiment a vibrating plane is used to generate wrinkles in the water .if the distance between two successive point is 3.5cm and the wave travel a distance of 31.5cm find the frequency of the vibration
Tamdy
the range of objects and phenomena studied in physics is
what is Linear motion
straight line motion is called linear motion
then what
Amera
linear motion is a motion in a line, be it in a straight line or in a non straight line. It is the rate of change of distance.
Saeedul
Hi
aliyu
Richard
Linear motion is a one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension
Jason
is a one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimensions.
Praise
what is a classical electrodynamics?
Marga
what is dynamics
Marga
dynamic is the force that stimulates change or progress within the system or process
Oze
what is the formula to calculate wavelength of the incident light
if a spring is is stiffness of 950nm-1 what work will be done in extending the spring by 60mmp
State the forms of energy
machanical
Ridwan
Word : Mechanical wave Definition : The waves, which need a material medium for their propagation, e.g., Sound waves. \n\nOther Definition: The waves, which need a material medium for their propagation, are called mechanical waves. Mechanical waves are also called elastic waves. Sound waves, water waves are examples of mechanical waves.t Definition: wave consisting of periodic motion of matter; e.g. sound wave or water wave as opposed to electromagnetic wave.h
correct
Akinpelu
what is mechanical wave
a wave which require material medium for its propagation
syed
The S.I unit for power is what?
watt
Okoli
Am I correct
Okoli
it can be in kilowatt, megawatt and so
Femi
yes
Femi
correct
Jaheim
kW
Akinpelu
OK that's right
Samuel
SI.unit of power is.watt=j/c.but kw.and Mw are bigger.umots
syed
What is physics
study of matter and its nature
Akinpelu
The word physics comes from a Greek word Physicos which means Nature.The Knowledge of Nature. It is branch of science which deals with the matter and energy and interaction between them.
Uniform
why in circular motion, a tangential acceleration can change the magnitude of the velocity but not its direction
reasonable
Femi
because it is balanced by the inward acceleration otherwise known as centripetal acceleration
MUSTAPHA
What is a wave
Tramsmission of energy through a media
Mateo
is the disturbance that carry materials as propagation from one medium to another
Akinpelu
mistakes thanks
Akinpelu
find the triple product of (A*B).C given that A =i + 4j, B=2i - 3j and C = i + k
Difference between north seeking pole and south seeking pole