# Range of Physics Week Day 3: Angular Momentum on an Astronomical Scale

Yestarday we saw that a cycle does not fall because of the rotation

of its wheels

Today we will see some places where MASSIVE objects/phenomenon is caused/affected by rotations and thus angular momentum .

a) Accretion Disks in Binary Systems : Our Sun is a single star . But most

of the stars in the Universe occur in Pairs . They form what is known as

Binary systems . Now lets suppose that there are 2 stars in a binary

pair : A HUGE star with a low atmosphere density . Lets call this dude S

and we have a compact star s which is the companion of S . If s and S are

close to each other then it might happen that s may begin to attract

the gases that form the atmosphere of S . so the gases of S start moving

towards s . Now as they move they loose energy because of collisions .

So to conserve the Angular Momentum the gases start whirling around

s and form a disk/ring around s . This is called the accretion disk .

The ring is flat and rotates very fast . The ring is perpendicular to

the axis of rotation of the star s . This is because the gases at this

place experience more centrifugal force and can resist the gravity of s .

This is again in accordance with the consevation of Angular Momentum .

For a beautiful picture of this Phenomenon see :

http://physics.angelo.edu/~msonntag/physics1301/binarymasstransfer.htm

b) Formation of Galaxies :

Imagine : A HUUUUUUUUUUGE cloud of dust and gas spinning . IT collapses

because of the gravity of its individual parts or some local irregularities.

Now as mass falls towards the inside it starts rotating very rapidly.

THis is to conserve angular momentum . Again it collects in a disk shape

perpendicular to the axis of rotation . Now places where there is more

gas/dust mass start falling withn themselves to form stars . Again these

starts are rotating . The total angular momentum of the galaxy is

the sum of the rotational motion of the galaxy and the sum of angular

momentum of its parts( starts , dust etc ) .

So we see that the same theory that explains tops also explains

things zillions of miles across .

# Range of Physics Week Day 5: Angular Momentum in Quantum Mechanics: Electron Spin

Today we come to the world of the ultrasmall . The world of electrons , photons and other elementary particles .

And the world here is very different fom anything we can ever imagine . The most important things here are probabilities , uncertainties and quantizations . How does this Change our view of Angular Momentum .

Most important thing : Angular momentum is still conserved .

However the way we imagine it becomes very different . Here is how it

is ( The actual theory is very vast. But here are a few glimpses) :

Lets take an electron : Every particle has a spin quantum number.

It has to be either integral or half integral . For an electron the

spin quantum number is 1/2 ( half ) . This means that en electron

can have the following values of its spin Angular Momentum( z component

actually ) :

+1/2 * h/2*pi          or           -1/2 * h/2*pi

An electron ( any electron ) can exist with just these 2 values of

angular momentum . You cannot make a electron have more angular

momentum or less angular momentum by making it spin faster or slower

or doing anything of this sort . So we see a wierd thing here .

Angular momentum becomes quantized . Similarly a particle with a spin

value of 1 can have the following 3 values of angular momentum :

+1 * h/2 * pi    or      0           or      -1 * h/2 * pi

But if you put 2 electrons in a system the total spin of the system

becomes 1 and it now behaves just like a spin 1 particle ( ie has 3

values of angular momentum ) .

Particles with half integral spins ( like electrons ) are called

fermions and no 2 of them can be in the same state . This is pauli’s

exclusion principle .

Particles with integral  spins ( like photons ) tend to get together

and are called bosons.

Now the term Electron spin is very misleading . NOBODY has ever

seen a electron spin around its axis . Electon does not even have a axis .

And you cant even see a electon spin around its axis if it did ( would

violate the uncertainty principle ) . So final word : electron does not

spin !!! The spin quantum number is actually means that the electron

has angular momentum even without rotating ðŸ™‚ . cool aint it .

So has this spoilt the mood with which we started the week ?

I said that I will take a theory and show you how it applies at

all scales to the Universe . And now we see that our theory has to

change to suit the atomic world . Now let me say that the theory that

we applied to galaxies and cycles was a special case of the real

correct theory of Angular Momentum . And the correct thoery is the

theory that applies to elementary particles . So this theory ( Quantum Mechanics) is the RIGHT theory and all the stuff we studied in schools( Newtons

laws , electricity , magnetism etc ) can be dereived from Quantim Mechanics .

So the theory of Angular momentum does apply to all range in the

Universe . It is just that at different orders of Magnitude the effects

that become important( and dominate) are different .

God created the fabric of the Universe from a single thread !

# Range of Physics Week: day 4 Angular Momentum on a Earth Scale: Coriolis Force

Yesterday we discussed about how Angular Momentum and its conservation produces some effects on an Astronomical scale .

Today we will talk about a effect it causes on Earth. It is called coriolis force .

Imagine a rotating flat disk and you are standing on its end .

Let your mass be m and you be standing at a distance R from the axis

of rotation . Let the disk be rotating at an angular velocity of w .

So your contribution to the angular momentum is m * (R^2) * w . the

m * R ^ 2 is your moment of inertia .

Suppose now to try and move towards the axis of rotation .

Now your distance from the axis be r . Now your new angular momentum

is m * ( r ^ 2 ) * w . It has become less . But it cannot just become

less . It has to be conserved . So you will experience a sideways force !

So a body on a rotating thing , when it moves towards/away from the

axis experiences a sideways force . This is called the Coriolis Force .

So if you move towards the centre of a Merry go around u will

get a sideways kick ( depending on the speed of the merry go around and

ur speed ) .

Now our Earth is also a rotating system and the Coriolis force

produces some interesting effects here too .

A projectile when moving on the Northern Hemisphere of earth

experiences a deflection towards its right . Similarly in the southern

hemisphere there is a deflection towards the left . Missile designers

have to take care of this because the deflection can be very big for long

range missiles .

Winds on the surface of earth move from high pressure to low

pressure areas . If earth were not rotating the winds would be parallel to

isobars . But earth ghoom rahi hai ! so the coriolis force comes and the

winds are deflected to the right on the Northern Hemisphere !

Coriolis force is a very visible phenomenon . River beds are dug

deeper on one side the on the other( right side on the Northern Hemisphere) .

Coriolis force has also resulted in a lot of hype . People claim

that water in the sinks drain/move counterclockwise in the Northern

Hemishere and clockwise in Southern Hemisphere . But this is not because

of the Coriolis force . The flush/sink designers make it look that way !

Coriolis force cannot deflect the fast moving water in a small sink .

The sink would have to be hundreds of metres in radius to produce a

visible effect . Could we say that here we have an application of

Angular Momentum in Commerce !!! ðŸ™‚

We even have a poem about Coriolis force :

May the Force be with you

————————–

On a merry-go-round in the night,

Coriolis was shaken with fright.

Despite how he walked,

‘Twas like he was stalked,

By some fiend always pushing him right.

kyon ? ghumaa diya naaa ? ðŸ˜‰

# Range of Physics Week: Day 2 : What is Angular Momentum

Most of us would probably be familiar with the basic theory of angular momentum . But for completeness sake here it is in a very very abridged format.

Angular momentum is a measure of a body’s tendency to continue to spin around its axis or revolve around something else . It depends on

2 things :

1) The distribution  of the mass in the body . A body whose mass is close to

the axis is more diffcult to start and stop rotating. A body whose mass

is far away from the axis is easier to rotate / stop rotating .

Also for the same distribution the heavier body is more difficult to

stop/start rotating .

This factor is called Moment of Inertia . Its symbol is I .

This is the counterpart of mass in linear motion .

2. The angular velocity of the body : You would appreciate that it is

easier to make slowly rotating body stop and more difficult to

make a rapidly rotating body stop . The angular velocity of a body

is the speed( velocity to be exact ) of its rotational motion . Angle

covered in unit time . Lets call this factor w

So we define angular momentum as I w

But we also need a direction for this quantity . Why ? To differentiate

between say clockwise and anticlockwise rotations . So some people defined

a direction of motion to perpendicular to the plane of the rotation of

the body and the direction is given by the “right hand rule” .

If you wrap your fingers in the direction of the rotations then the

the direction of the thumb gives you the direction of the angular momentum.

For example if u stand and look at the clock then the direction of

angular momentum of the second hand is right into the clock ( at right

angles) .

Now another important thing about angular momentum is that if

you want to change it you have to apply an external force ( torque) to

it . The angular momentum of a system left alone( and of the whole world)

remains constant . That means that angular momentum is a conserved

quantity.

Even if you want to change the direction of motion you still have to

apply an external torque .

Why does a moving bicycle not fall ? The reason is that to make it fall

we will have to change the angular momentum of the wheels and to do

that we will have to apply a large external force ( at the right place ).

So it is the rotation of the wheels of a bicycle that give it stability.

The faster it moves the more stable it is .

That is why a top keeps rotating and does not fall . That is why

bullets that spin on the way to the target have more stability

than those that dont . That is why all rockets/satellites are deliberately

Rotation gives the body stability . It takes a large torque to change

the direction of rotation of a rapidly rotating body . So it is more stable.

# Range of Physics Week: Day 1 : Some “Philosophical” Remarks

First lets examine the question : Why is there so much rotation

in the Universe . Any star , any planet , any galaxy , any anything

rotates . Why is that ?

Let me answer that by sating that this is a bad question .

The reason is that rotation is a natural degree of freedom and

you cannot ask a body ” why do you rotate?” . If you go someplace

in the Universe and a body is rotating u should not ask Why .

If you go someplace and some body is not rotating Then u SHOULD

ASK : “why are you not rotating ? ” Because it is strange if

a freely suspended body does not rotate. Not rotating is strange ,

rotating is not strange .

If a body is not rotating that means that there is a perfect

balance of forces(torques) from all sides . That is usually

not possible ( the probability is very small ) . And a body

if it starts rotating , then it is very difficult to make it

stop . You have to apply the right amount of force and the right

place and only then it will stops . And if by mistake you

apply a little bit extra or a little bit less force or

force at a little bit different angle then all your efforts are

useless because the body will not stop rotating . Just the

nature of the rotation will change . So the probabilty that a

random force will start a rotation is VERY LARGE but the

probability that a random force will stop the rotation is VERY SMALL .

So we se that there is a HUGE number of things in the UNiverse

which keep rotating . Things made of all sorts of constiuents,

things rotating in all sorts of wierd ways . But the amazing thing

about Physics is that all rotations , of any body is explained

by the Theowy of Angular Momentum . One concept( Angular Momentum)

describes all rotations .

Some people feel that there is nothing great about it  .

A ball rotates , a electron rotates . So if u treat the electron

just like a ball then the same theory should apply .

But notice that this is wrong . Lets examine the rotation of

a ball . It consists of a huge amount of smaller pieces and when

we say that a ball rotates it actually means ” The pieces away

from the axis are revolving around the axis ” . So in a sense

there is a revolution and not a individual piece rotation .

And each of these pieces are in turn made of smaller pieces .

Have you ever seen a INDIVIDUAL PIECE spin ???? No .

In the observable world around us there is nothing like a

individual piece . Everyhing we see around us has millions and

zillions of pieces in it . So a theory od spin for a cricket ball

applies to a piece made of millions of smaller pieces .

But an electron is a individual piece ! In the truest sense of

the term . So should the same cricket ball spin theory apply to

it ???????

So it is not obvious that the theory of angular momentum that we

made of a cricket ball should apply to the electron as well .

But it does apply . And there is the power of the ideas of Physics !

We will see later that we do need some modifications to

apply the same theory to the electron and the ball . ðŸ™‚

This is why I want to demonstrate the scalability in Physics .

That this kind of a thing is not obvious . But still it happens .

And that is what makes Physics so fascinating .

# Range of Physics Week

Hi ,

For the next five days we are going to talk about Angular momentum .

The aim is more sentimental than academic . I want to demonstrate

the range at which a basic concept of Physics is valid .

So we are going to take a concept of Physics( Angular Momentum )

and take it to all scales : The scale of the Universe ( 10 ^ 15 metres),

then we will take the same concept to a earth scale( 10 ^ 2 metres )

and then we will take the same principle into the depths of

subatomic particles( 10 ^ ( -15 ) metres ) . So you will be

able to see that a single law of Physics can be applicable for

30 orders of magnitude !!! We will see how it changes its form

at various scales and what parts remain the same .

If you take a step back and see what it means you would feel

a sense of wonder . Do you know of ANY other subject/idea

apart from Physics that has such a Universal validity ?

You take it anywhere and it still works !

Einstein said ” Everything else is for the time being !

But an Equation … An Equation is something for Eternity “

The agenda for the next five days :

Day 1 12 June : Curtain raiser , some general/philosophical questions.

Day 2 13 June : What is Angular Momentum . Definitions .

Day 3 14 June : Cosmological scale : Galaxy Formation, Accretion Discs Day 4

15 June : Earth Scale : Coriolis Force and its effects Day 5

16 June : Subatomic Scale : Angular momentum in Quantum Mechanics