# Conservation of angular momentum

• zezima1
In summary,Everyone knows that an ice skater can increase her speed by pulling her arms in. Doing so the moment of inertia decreases, and thus her angular velocity MUST increase due to the conservation of angular momentum. However, work is also done in the process. My question is: if she then pulls her arms out again, will she spin slower again? I think so, because angular momentum is conserved. However, I am unable to see how she would do negative work on herself (which she MUST to lose the extra kinetic energy gained from pulling in her arms).

#### zezima1

Everyone knows how an ice-skater can increase her speed by pulling her arms in. Doing so the moment of inertia decreases, and thus her angular velocity MUST increase due to the conservation of angular momentum. But work is also done in the process. My question is:
Say the ice skater pulls in her arms and spins faster. If she then pulls the out again: Will she spin slower again? I think so yes, because angular momentum is conserved. I am however unable to see how she would do negative work on herself (which she MUST to lose the extra kinetic energy gained from pulling in her arms).

hi zezima1!
zezima1 said:
If she then pulls the out again … I am however unable to see how she would do negative work on herself (which she MUST to lose the extra kinetic energy gained from pulling in her arms).

she does negative work on her arms

if her arms were on strings, they would naturally fly out to the full extent of the strings

to pull them in, she needs to do work … the tension is in the same direction as the displacement

to let them out again, she just let's the strings out … so the work done is negative … the tension is in the opposite direction to the displacement

but since her angular velocity is changing there should be a torque .and obviously the torque of tension is zero wrt the axis.

pcm said:
but since her angular velocity is changing there should be a torque

no, you only need a torque to change angular momentum

(and it doesn't change)

oops...
sorry for that,feeling sleepy..

but still we need a tangential force to change angular velocity.
how can only a radial force change angular velocity.

pcm said:
but still we need a tangential force to change angular velocity.
how can only a radial force change angular velocity.

still sleepy! :zzz:

force only changes momentum (or angular momentum)

Hi tim, thanks for the answer. I'm still finding it a bit hard to swallow though. In the case, where she pulls in her arms she does a positive work on them. This extra energy must come from chemical energy in her cells. In the case where she pulls out her arms, where does the energy go? Certainly it doesn't go back into the cells!

aaaa202 said:
where she pulls out her arms, where does the energy go? Certainly it doesn't go back into the cells!

she can store it in springs, if she wants to

if she doesn't, it's like where does the energy go when she walks downstairs …

if she falls downstairs, it goes into kinetic energy, but if she controls her speed, her cells have to use up energy to do so

that's the way the body works, it's very wasteful even when it's doing no work

the problem rephrased:
no net external torque ,implies L is conserved.
she does work in pulling her arms,implies KE changes.
but now consider her inner body(excluding arms hands) as our system,
now since in the whole process moment of inertia of inner body does not change and angular velocity increases, there is an increase in angular momentum of her inner body.
but the force from outside the system is that from her arms (and also normal force,weight etc.) and the torque of that is zero.so what torque produces change in L of her inner body.

Last edited:
pcm said:
… but the force from outside the system is that from her arms (and also normal force,weight etc.) and the torque of that is zero.so what torque produces change in L of her inner body.

oooh, that's interesting … I've never seen that mentioned before!

yes, in addition to the radial force needed to pull the arms in, which does work, and which everyone mentions

there's a tangential force needed, which of course does no work (because it's perpendicular to the displacement), and so no-one ever bothers to mention it!

this force provides a torque which reduces the angular momentum of the arms,

and the equal and opposite reaction torque increases the angular momentum of the inner body

well spotted!

so what is that tangential force?
i can not find one.

when she pulls her arms in,

she also has to pull them back slightly, to reduce their angular momentum

## What is conservation of angular momentum?

Conservation of angular momentum is a fundamental law of physics that states that the total angular momentum of a closed system remains constant over time, as long as there is no external torque acting on the system.

## Why is conservation of angular momentum important?

Conservation of angular momentum is important because it helps explain and predict the motion of objects in rotational motion. It is also a fundamental principle that is used in many areas of physics, such as astrophysics, mechanics, and fluid dynamics.

## How is angular momentum conserved?

Angular momentum is conserved when there is no net external torque acting on a system. This means that the total angular momentum before an event or interaction is equal to the total angular momentum after the event or interaction.

## What are some real-life examples of conservation of angular momentum?

One example of conservation of angular momentum is the motion of a spinning top. As long as there is no external force acting on the top, its angular momentum will remain constant. Another example is the motion of a spinning ice skater, who can increase or decrease their speed by changing the distribution of their mass while conserving their angular momentum.

## How does conservation of angular momentum relate to the conservation of energy?

Conservation of angular momentum is closely related to the conservation of energy. Both are fundamental laws of physics that govern the behavior of closed systems. In fact, conservation of angular momentum is often used to solve problems involving rotational motion, which is a form of kinetic energy.