Deriving Newton:

Calculus Behind the 3 laws

 

Alexander Kareev

3rd hour

APPC

05/14/06

I certify that have neither given nor received unauthorized assistance on this test or assignment .

  

 

( A picture of me with guitar, Alex Kareev. Student of Fayetteville High school 12th grade and student of APPC 3rd  hour)

 

 

This web page was created as part of the curriculum for the AP Physics C (APPC) class. I am a student finishing this class and I decided to cover this topic of Calculus and the 3 laws of Newton as the web page topic.  You can contact me at A_KAREEV@FAYAR.net  and you can contact my teacher David A. Young at dyoung7@prodigy.net and or dyoung@fayar.net  .

 

 

 

Calculus is very important for the AP Physics C course. Derivation and Integration of equations and graphs is a very important part of this course but there is also integration of formulas   which is also important for physics. Being able to integrate and derive formulas and understand the relation between formulas will help you understand the relations between physical concepts such as force and energy for example .

 

            This is a very big topic and I will not go to far in it and neither will I try to explain the integration and derivation, I will just use it in context of Newton’s 3 laws to show how it work and also explain their importance. 

 

If you are conf used about calculus applied to formulas I will just give you a very popular derivation of formula

Her for example is a formula for Kinetic energy

 

KE= ½ mv^2

 

What happens if we teak a derivative of it (assume that m is a constant)

 

= ½ *2 mv  = m *v

 

Which is the momentum

 

P= mv

 

So the instantaneous slope of the kinetic energy graph will give as momentum. And area underneath the momentum graph will tell as the displacement in kinetic energy.

 

You see how this is connecting the physics concepts using math.

 

Lets for example say there is a 1 kg ball and it is moving with changing velocity here is the data

 

v (m/s)

KE (1/2 mv^2) (J)

0

0

1

0.5

2

2

3

4.5

4

8

5

12.5

  6

18

7

24.5

8

32

9

40.5

10

50

 

 

The instantaneous slope on each point would provide the momentum of the object

 

 

 

In the moment you will see how the 3 laws on Newton and their derivations relate to physics on a grand scale, and what impact he and the calculus that he invented had on physics.

 

 

Newton is famous for his 3 laws and law of gravity as well as for inventing calculus (the derivation part the integration part was invented by Leibniz). Newton also had a big impact on optics but his work in this area is not as well famed, and unfortunately not discussed here.

 

 

A brief history of Newton and derivation is brief because it starts at Newton,  he invented derivation. After he came up with his laws especially the F=MA he needed to find the acceleration ant to do this and for his convince he invented derivation, and actually kept it secret for a while.  

 

 

 

And here are some fundamental information about the calculus and Newton’s laws

 

You probably know the lame definition of the laws but here is how Newton and other physicist so it and why it is important.

 

 

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Newton's Laws of Motion are the foundations of classical physics, and of the study known as dynamics. In formulating his laws, Newton developed several concepts that have become vital to modern physics. In order, his laws are:

  1. Inertia -An object in motion will stay in motion and an object at rest will stay at rest, unless acted on by an unbalanced force. A body is in equilibrium when the net force acting on it is zero, this means that it's vector sum is zero. This law is only applies to inertial reference frames.
  2. F = ma, where F equals the sum of forces acting on a body, m is the mass of the object, and a is acceleration. Force and acceleration are both vector quantities and have the same direction. Acceleration is proportional to force but inversly proportional to mass which you can see by solving this equation for acceleration a = F/m
  3. Every action has an equal and opposite reaction. When two bodies interact they exert forces on eachother that are equal and opposite in magnitude and direction.

Inertia

Inertia is based on a simple concept: an object in motion tends to remain in motion. An object at rest tends to remain at rest. This is counter-intuitive to us, since we know that if we slide an object along the floor, it will slow down. Similarly, something may decide to fall off a shelf in a way that looks like it decided "all by itself". But Newton understood the basis behind this physical reality. When you slide a book along the floor it slows down, but not because of, as was sometimes thought, some property of matter that makes objects want to slow down. The book slows down because it interacts with the floor. Although this may seem a trivial difference from the way most people think about motion, it is really quite significant.

The concept of inertia refers to the tendency of an object to remain in motion, or to remain at rest. By intuition we can see that the tendency is somehow related to mass. It is easy to catch and stop a baseball; I don't recommend attempting to catch and stop a freight train. It's also a lot harder to move a loaded filing cabinet than it is to slide a single book. In order to measure how hard it was to overcome an object's inertia, Newton invented another concept:

[edit]

Force

Force can be a confusing term because it is an everyday term, as well as a physics term. When we use it in everyday language, we use it in a whole number of different contexts, for instance "He forced the door", "He was forced to take third semester Calculus", or "This justifies the use of force". There are a great many ways in which the word force may be used in casual conversation.

However, in physics, there is only one meaning to force, that which was given to us by Newton's Second Law. For now we will treat force as having been defined by:

\mathbf{F} = m\mathbf{a}


where m represents the mass of the object in question and a its acceleration.

When you consider Newton's first law, you have to wonder how an object starts moving. Something has to happen to an object that is standing still to make it move. What happens is that an external object applies some sort of force, which can be summed up as a push or a pull. We know that, of course. Ever since we started observing the natural world around us we have been aware of the way in which things are moved around us.

What is important about this law is that force is now quantified, and can be shared between two objects. For instance I can design a weight that applies a force of thirty Newtons to the end of a lever. I can use this to calculate the force I apply to the other end of the lever. Using force it will become possible to solve complex physical systems for important variables.

 

[edit]

Equal and Opposite Reactions

Now that we have the concept of force, we can begin exploring how it effects the world around us. Consider this. You are most probably sitting in a chair. How many forces are acting on you at this moment? Since you are not accelerating in any direction it would be easy to think that there are no forces, but that would be incorrect. Instead there are a great many forces that are all balanced. For instance, there is the force of gravity pulling you down. If you were magically suspended there above the Earth, the force of gravity would pull you swiftly, and ungraciously, to the ground. But you do not fall to the ground, instead you exert a force on the seat of the chair.

Here is where Newton's Third Law comes into play. You are exerting a force on the chair equal to your weight (let us assume for a moment that all your weight is resting on the chair). This is equal to:

\mathbf{F} = m\mathbf{g}


This is the force that you apply to the seat of your chair. However, thanks to Newton's Third Law, this is also the force that the chair applies upward on you. Because of this, the force of gravity pulling you down is countered by the force of the chair pushing you up, and you do not go sprawling all over the floor.

This is a hard concept, because it always applies, even when you might think it does not. Let us deal with a human attempting to catch a freight train. When the freight train hits the human, which has more force applied to it? The answer is, from the title of this section, that the forces are equal. The force applied to the human by the train is the same as the force applied to the train by the human. Now, you should note that a few thousand Newtons of force has a much greater effect on a human than it does on a barreling freight train, but the force is the same. It is the objects that are different, a trick that is often used to fool students on examinations.

 

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( the above information was copied from the http://en.wikibooks.org/wiki/Physics_with_Calculus:_Part_I:_Newton's_Laws

Website. I would like to rephrase it, but it is preaty compact and to the point. The site again is http://en.wikibooks.org/wiki/Physics_with_Calculus:_Part_I:_Newton's_Laws

. Go their to get the full picture)

 

 

 

 

 

 

Useful links :

http://home.case.edu/~sjr16/pre20th_europe_newton.html

http://en.wikipedia.org/wiki/Isaac_Newton

http://scienceworld.wolfram.com/biography/Newton.html

http://euler.ciens.ucv.ve/English/mathematics/newton.html

http://library.thinkquest.org/11902/physics/newton.html

http://www.niehs.nih.gov/kids/rdpartytx.htm

 

 

and the best page of all on this topic is:

 

http://en.wikibooks.org/wiki/Physics_with_Calculus:_Part_I:_Newton's_Laws