The Equations thread

[NP-complete]

The idea of this thread is that members will post an equation and explain it. This way we can benefit from the scientific/economic/financial knowledge of forum members.


We begin with Newton’s second law, F = ma

Where,
F stands for force
m stands for mass
a stands for acceleration

So it says force equals mass times acceleration.

To begin with lets first lets replace the ‘a’ in this equation with v/t because we know that change in velocity divided by time gives acceleration, we get

F = m(v/t)

or we can rewrite as F =(mv)/t

Now we need to understand a term that Newton used ‘quantity of motion’. The idea is simple. A moving body has motion in it. We want to ask how much motion a moving body has? We simply say a faster body has more motion in it than a slower body. Hence the quantity of motion depends upon the velocity of the body. But it also depends upon another thing, which is its mass.

To see how consider balls on a pool table. A moving ball strikes a stationary ball on the table and that ball starts moving. The moving ball transferred some motion to the stationary ball. Now we use for the moving ball a ball that is twice in size. This new ball moves with the same velocity and strikes the stationary ball. But this time the stationary ball moves off with twice the velocity as compared to the previous hit by the smaller ball. The motion transferred to the stationary ball by the bigger ball is twice the motion transferred by the smaller ball. We conclude that the quantity of motion in the bigger ball was twice the quantity of motion in the smaller ball. The velocity of both balls was the same. But doubling the size doubled the quantity of motion. Hence we conclude that quantity of motion also depends on the mass. Therefore quantity of motion is equal to mass times velocity or ‘mv’.

Force is an agent that moves a stationary body or speed up a moving body or slow down a moving body. Force changes the quantity of motion in a body. The greater the force applied to a body, the greater the rate of change of quantity of motion in that body. That’s what Newton’s second law says: “Force is the rate of change of quantity of motion of a body.”

F = change in quantity of motion / time

Or, F = (mv)/t

Or, F = ma

 

[war&peace]

The equation for mass conservation is as follows

div (rho * U) = 0 ​

This is basically for compressible flow but caters for incompressible flows also since I call the incompressible flow to be a special case of compressible flow. The density of the fluid changes in a compressible flow and if it does not change, means the flow is incompressible (i.e. it cannot be compressed or conversely, the density remains unchanged). The mass conservations tells us that under a non-nuclear and non-reacting conditions, the mass of the system remains constant.

The div operator means divergence which i.e. the partial derivatives w.r.t to coordinates, in cartesian coordinates, for a 3D flow case, it will x, y, z, in polar coordinates, r, theta, phi and in cylindrical coordinates, it will be x, r, phi.
In tensorial form we can used x1, x2, x3 but I guess the tensor may be a bit more involved concept so I would skip it for time being.

Since we talk a lot about airplanes on this forum, lets analysis the basic mechanics of the flight. An Airplane to fly with constant speed and a levelled flight needs to generate thrust force (T) equal to to the resistance it faces, called Drag (D) and to maintain the level or altitude, it needs to lift is weight (W) by generating a lift force (L)
so we have

T = D​

and

L = W​

Now during taxi T>D, and take off and climb L>W, in cruise all forces are equal i.e. balanced, during landing L<W and after touch down D>T

Once we have understood these concepts, we can then relate these to the control surfaces of the airplane to get the real picture

​

 

[Levina]

I want to explain an equation which finds its application in our daily life,

P= F/A
where,
P= Pressure
F= Force
A= Area

Pressure is inversely proportional to area, in short pressure increases when the area of application of force is reduced.
For example,
My physics professor used to say girls exert more pressure on the surface of earth as they wear high heel sandals. How so?
High heel sandals have a tiny area of contact with surface and ergo the pressure exerted by a sandal is high.
(Ofcourse this was a joke. )
There are few more examples like
1) while watering plants in your garden if you were to put your finger on the mouth of hose pipe, then the pressure with which water comes out of the hose increases.
2)If you are peeling an apple, then if the knife is sharp and the area of contact is small then you can peel the apple with less force on knife.
3) If you must get an injection, then it is better to have a sharp needle than a dull one since the smaller area of contact means lesser force is required to push the needle through the skin.

@jamahir @Skull and Bones @Slav Defence @anant_s @thesolar65
I hope you guys will contribute to this thread.

 

[war&peace]

My physics professor used to say girls exert more pressure on the surface of earth as they wear high heel sandals. How so?
High heel sandals have a tiny area of contact with surface and ergo the pressure exerted by the sandal is high.
(Ofcourse this was a joke. )

Well professors are funny. According to my teacher (A professor in aerothermodynamics), the girls are more aerodynamic than men but I think the explanation is out of the scope of this thread.

 

[Levina]

The equation for mass conservation is as follows

div (rho * U) = 0 ​

This is basically for compressible flow but caters for incompressible flows also since I call the incompressible flow to be a special case of compressible flow. The density of the fluid changes in a compressible flow and if it does not change, means the flow is incompressible (i.e. it cannot be compressed or conversely, the density remains unchanged). The mass conservations tells us that under a non-nuclear and non-reacting conditions, the mass of the system remains constant.

The div operator means divergence which i.e. the partial derivatives w.r.t to coordinates, in cartesian coordinates, for a 3D flow case, it will x, y, z, in polar coordinates, r, theta, phi and in cylindrical coordinates, it will be x, r, phi.
In tensorial form we can used x1, x2, x3 but I guess the tensor may be a bit more involved concept so I would skip it for time being.

Since we talk a lot about airplanes on this forum, lets analysis the basic mechanics of the flight. An Airplane to fly with constant speed and a levelled flight needs to generate thrust force (T) equal to to the resistance it faces, called Drag (D) and to maintain the level or altitude, it needs to lift is weight (W) by generating a lift force (L)
so we have

T = D​

and

L = W​

Now during taxi T>D, and take off and climb L>W, in cruise all forces are equal i.e. balanced, during landing L<W and after touch down D>T

Once we have understood these concepts, we can then relate these to the control surfaces of the airplane to get the real picture

​

Gosh!
I remember very little from fluid mechanics. Drag was something that I read back in college.

Well professors are funny. According to my teacher (A professor in aerothermodynamics), the girls are more aerodynamic than men but I think the explanation is out of the scope of this thread.

Hahaha
Aerodynamic?
How so?
Let me guess they have lesser drag, because they have more curves???
Lolzzz
Science teachers can be really funny at times.

 

[Whazzup]

Gosh!
I remember very little from fluid mechabics. Drag was something that I read back in college.

Should have concentrated more in second year. Fluid mechanics was the easiest , DSS was tough if u have no interest in it.

 

[war&peace]

Gosh!
I remember very little from fluid mechabics. Drag was something that I read back in college.

I think the main of this thread is to provide an introduction to science for newbies and a refresher for others. I have some background in fluid mechanics and flight mechanics. Lets develop this thread since it is non political and good for every one.

 

[Levina]

Should have concentrated more in second year. Fluid mechanics was the easiest , DSS was tough if u have no interest in it.

fluid mechanics is an interesting subject. But I don't remember the equations. And I have been out of touch with the subject for long now.

I think the main of this thread is to provide an introduction to science for newbies and a refresher for others. I have some background in fluid mechanics and flight mechanics. Lets develop this thread since it is non political and good for every one.

I would not have enjoyed science if my professors had not made it interesting.
I seriously believe that some element of fun helps you understand science better.
So I hope this thread doesn't turn very serious.

 

[Skull and Bones]

I intend to work in the field of micro-fluidics for my device prototype.

 

[Levina]

I intend to work in the field of micro-fluidics for my device prototype.

Uncle,
I do not understand your PhD research topic. Could you pls explain a simple science equation? Just about any equation?

****
Kidding!
I just hope you'll contribute to this thread.

 

[war&peace]

Okay, I should get in my funny style and sugar coat the fluid mechanics pill.

Lets start where we left, I have already discussed the force balance in comment # 2, now lets delve little bit into the details of each one. But first determine the good and bad guys of these equations. Thrust and Lift are hero and heroine, respectively, in this sequel while Drag and Weight are the villains for an airplane. Drag is like a dragon that is trying to stop the airplane and that is why he has to drink more fuel to produce more Thrust to overcome this dragon ( technically drag is the resistance of the fluid to any object (our little airplane) moving through it.

 

[jamahir]

For example,
My physics professor used to say girls exert more pressure on the surface of earth as they wear high heel sandals. How so?
High heel sandals have a tiny area of contact with surface and ergo the pressure exerted by the sandal is high.

hah !!

the good professor was ideally correct but technically wrong for some cases... size-zero twiggies in heels don't exert pressure.

of course, i have not been trampled by a bunch of them so i am not sure.

There are few more examples like
1) while watering plants in your garden if you were to put your finger on the mouth of hose pipe, then the pressure with which water comes out of the hose increases.
2)If you are peeling an apple, then if the knife is sharp and the area of contact is small then you can peel the apple with less force on knife.
3) If you must get an injection, then it is better to have a sharp needle than a dull one since the smaller area of contact means lesser force is required to push the needle through the skin.

good examples from everyday life.

I hope you guys will contribute to this thread.

my computer programmer mind will take some time to read the op and war/peace's post.

 

[Skull and Bones]

Uncle,
I do not understand your PhD research topic. Could you pls explain a simple science equation? Just about any equation?

****
Kidding!
I just hope you'll contribute to this thread.

I'm working in the field of molecular electronics, and my research has a lot to do with zero energy molecular sensors and i deal with tunneling of electrons through the potential barriers in sub 10 nano meter distance.

Where the current to distance of electrode relation is given by,

 

[war&peace]

I'm working in the field of molecular electronics, and my research has a lot to do with zero energy molecular sensors and i deal with tunneling of electrons through the potential barriers in sub 10 nano meter distance.

Where the current to distance of electrode relation is given by,

Then it is nano-fluids. I worked with microfluids. The micro world is ruled by the queen Viscosity and Mr Inertia has not much say in this matter. At nano scales even the quantum effects play their role.

 

[Levina]

Now during taxi T>D, and take off and climb L>W, in cruise all forces are equal i.e. balanced, during landing L<W and after touch down D>T

That's how the forces are balanced?
So the total force = lift + drag
How do you fit everything in one equation???

I'm working in the field of molecular electronics, and my research has a lot to do with zero energy molecular sensors and i deal with tunneling of electrons through the potential barriers in sub 10 nano meter distance.

Where the current to distance of electrode relation is given by,

Yes you have told me about it, but I must say that you're such a show off.
Explain something that will help the newbies.