Overview

By using a spinning chair, weird effects due conservation of angular momentum can felt. Have a go before demonstrating, but beware - it's difficult to concentrate when your head is spinning!

Set up:

What can you do with it?

Masses:

• By holding the masses at different distances from the body, the speed of spinning changes. For small children, it's enough to get them to just stick their arms and legs out.

Wheels:

• Before sitting on the spinny chair, hold the bicycle wheel between two hards and spin it. Then your arms so that the wheel is horizontal. There is a pull in the opposite direction.
• Stopping the wheel while it's spinning. For a child standing, stop the wheel so that they feel a twist in their feet. For a child on the chair, if they are stationary, start the wheel spinning then stop it. They will start to spin.
• Turning the wheel over while sitting on the spinny chair (initially stationary) so they move in one direction then the other.

Other things to talk about

Artificial satellites use gyroscopes to orient themselves without wasting reaction mass.

Stable rotation axes - important for satellites.
Spinning in trampolining or ice-skating: to go faster, trampolinists and ice-skaters bring their arms and/or legs in.

Tips for Demonstrating:

Basic Procedure and Explanation:

"Who wants a go on my spinny chair?"

Ice Skater:

Give them the masses and set them spinning with a gentle push on the arm. Two masses, one in each hand, arms out sideways is most effective (add legs up forwards too, if the child is heavy)
One mass held in both hands, held out forwards works for lighter children, especially when complemented by legs out too.
For the smallest children don't use masses, just get them to stick their arms and legs out.Get them to experiment with
bringing the masses in to their bodies and away again (concentrating and distributing their mass). Miming the actions as you say them helps communication here.
Stop them spinning then ask first what happens when the masses are in different places, then ask why it happens.

Get them to observe the paths the masses are moving along (repeat experiment if necessary). The idea is: it is a
shorter distance around a small circle than a large one, while the masses want to keep the same speed, so get
around the smaller circle quicker.

Sadly, this is only half of what is happening here. Neglecting the mass of the child, the linear velocity
of the masses would actually increase as they are pulled in. This follows from the formlua for the angular
momentum of a particle: L = m v x r The mechanism for this is more subtle, we don't yet have a field tested
explanation for it. [To do? - perhaps something along the lines of conservation of energy: to bring the weights
in, you have to do work, which you can feel]

One way of thinking about momentum is loaded shopping trollies - to get them moving you have to give them a push,
and they won't stop moving unless you restrain them

To give them a feeling for how moment of inertia varies with mass distribution you can get on the chair yourself
with the two large masses, and get them to spin you slowly, first with concentrated mass, then distributed.
What happens to all the extra effort they put in to spin you in distributed configuration when you make yourself
concentrated? (True - the extra angular momentum they supplied is conserved, and you did start with more energy,
but you do work too when you pull the masses in) (Warning: it can be hard to demonstrate when your head is
spinning...)

Stopping the Wheel:

Sit a child on the chair, make sure they can hold the wheel (remember they also have to be able to support the
wheel when spinning. Be careful about abrasions to bare skin). Explain what you are about to ask them to hold the
wheel, you will start it spinning then will stop the wheel spinning. Then spin up the wheel, hold it level and
give it to them.

Repeat the experiment while standing on the ground and ask them to focus on how their feet feel as they transfer
the spin into the ground (depending on how credulous they are - explain that they are making the whole world spin,
just not very much.)

Back on the chair; you can't twist the world through the chair, and the spin won't disappear, so all you can do
is share out the spin between you and the wheel.

Turning the Wheel Over:

Spin the wheel slowly on one side, ask which way it is spinning, turn it over. Which way is it spinning now?
So we can change the way the wheel is spinning just by turning it over.

Get them on the chair, spin up the wheel and get them to hold it up-right, one hand on each side. Stand back
and let them experiment with turning the wheel (slowly!) onto one side and then the other. Observe directions
of rotation, and talk through how the angular momentum adds up to zero around the chair's free axis.

Where might this be useful? Artificial satalites can use gyroscopes to orientate themselves without wasting
reaction mass.

Background:

Angular momentum is an odd concept. The conservation of angular momentum can be derived from the symmetry of the
laws of physics with respect to rotations of the referance frame (c.f. classical mechanics). Or you can ignore
it and stick to point masses with negligible structure (or things that are built up by summing or integrating
over such masses). That way the symmetry will seep into your calculations through the geometry, and you will
still automatically conserve angular momentum.

It is only when you begin to leave out large chunks of a problem's geometry (treat an extended body as a 'thing'
rather than the sum of its parts) that you need to make angular momentum explicit in a calculation. Looking at
it more positively, angular momentum is the only aspect of the dynamics of the extended body that you need to
retain - so you can discard a lot of irrelevant detail at the start of a calculation rather than waiting for it
to cancel out at the end.)

It is the tendency of people's intuition to leave out geometry (to apply their experience of large non-rotating
objects to the wheel), but not replace it with a-m that makes this seem so weird.

My normal approach to the spinny chair is to begin by working in terms of point masses, showing how this gives
rise to angular momentum (although I shy away from the technical term, and just call it spin) - trying to get people
to add these ideas to their intuition. Then if they stick it out all the way through to the precessing gyroscope,
I try and pick that apart in terms of point masses and centripetal forces.

For the 'ice-skater' experiment, to bring the weights onto a smaller circle in the lab frame, you have to give them some radial velocity
in addition to the tangential velocity they already have. You don't get to take this velocity away again
when the weights reach the desired radius. To see this, consider applying the delta v as a single inward radial
impulse, then turn off the centripetal force until the desired radius is reached. The minimum impulse that
can be applied to achieve a given radius will make the mass's velocity tangent to a circle of that radius, so
that when the minimum radius is reached there is no radial velocity to reclaim.

In the original rotating frame, the particle begins by moving radially inward, but gets increasingly ahead of the
rotating frame (in the lab frame it is taking a short-cut across a chord of the circle), so its path curves in the
direction of rotation. This is the fictitious Coriolis force.

To move to a larger radius, first turn off the centripetal force, then apply the impulse once the new radius is
reached. This time the angle between the radial impulse and the particle's velocity is obtuse, so we can extract
energy from the particle. This is not straightforward to explain to children. You might be able to say something
about how, when the masses are spiralling inward, the inward pull is no longer purely sideways to their paths.
Also, they might accept the conservation of energy argument as they will have really been able to feel the work
they do to pull the masses in. This could be confusing though as in the person's instantaneous rest frame, radius
acts like height for potential energy.

[image2]

The Angular Momentum Version:

A-M is not constant, but its magnitude remains the same because the plane of the couple is always at right-angles to the current plane of rotation (because the . Say the wheel is spinning in the x-z plane, while the couple is in the y-z plane - the wheel precesses so that it is spinning in the y-z plane (you can only spin around one axis at a time). But the couple is still 90 degrees ahead... (or use axial vectors if you prefer.)

My Gyroscope Exlanation: [to do: do we still do this?]

Standing on the ground, let them hold the wheel with one hand on each handle.

(Be consistent in which way you spin the wheel for this part of the experiment - it is harder to get lost
in your explanation if you always do it the same way). Twist opposite handles of the wheel up and down,
feeling the pushes and pulls on the handles (in and out in alternate arms). Hold the handles level, and
rotate whole body from side to side (i.e. twisting the wheel sideways without bashing it against arms), again
feeling the forces (up and down in opposite hands).

Get their observations and help them summarise them while you thread the string through the end of one of
the handles. With the wheel not spinning, hold the string in one hand and keep the handles level with the other,
get them to predict, then release so that you are only holding the string. Describe this motion as a twist. Try for
predictions of what will happen when we try the same thing with a spinning wheel, then demonstrate it. Once released,
push the handle without string around in the direction of precession a little so that the wheel sits up and looks
even weirder.

The key to picking apart what is happening here is to realise that things can only spin round in one lot of circles
at a time, if you try make it something spin two ways at once its component particles are doing something much more
complicated.

Talk about how things move in straight lines unless they are being pulled sideways. Go back to shopping trollies,
conkers on strings, (drop blue-tac onto the rim and show how it flies off?).

Discuss how when wheel is rotating only one way, forces are all towards the middle, and all balance out,
so are not felt in the handles. Twirling a conker, forces don't balance. With the handles level, and the wheel
spinning top towards you while precessing around anti-clockwise (when looked down on), think about the motion of
its consitiuent particles, and the forces required to generate that motion (and thus supplied by the hands). The
bit of the wheel that is at the top would just come towards you, but as you are forcing the wheel to precess round,
you are making it curve sideways to your right. Likewise the bottom of the wheel needs a sideways force in the other
direction to make it curve to your left. These forces are opposite, but don't line up, so they are a twist. You can
feel your hands supplying this push as you force the wheel to precess, or you can hold the handles only at one end
and balance out the twist this generates with the precession force.

(This seems to add up correctly to a linear dependance of the force on the angular velocity of the wheel: as it goes
faster, the curves become straighter, but the mass flows around them faster, /and/ at a greater rate.) If drawing a
diagram of this, follow the path of the top of the wheel for a few more revolutions - projected on the horizontal
plane this will be a complicated epicyclic curve (see glowstick photo) - it is all the sideways curving that is generating the odd forces.

Nutation:

The more I think about nutation, the less sure I am that I understand it. This link shows you how complicated you can make it. At some point i intend to try and translate some of that into english... (Anyone with brain-cells to burn is welcome to try and beat me to it.)

http://mitpress.mit.edu/SICM/book-Z-H-40.html#%_sec_Temp_261