Visualizing Fractions

Article status:                              Draft
Time Estimate for Reading:        15 min
Learning Objectives:                   Fractions, Introductory Mathematics
Effort Required:                          low
Pedagogy Model:                        Collaborative Learning
Prior Math Tools:                        Counting numbers and basic arithmetic
Authors:                                       Rithanya, Srishti and Keerthana


Background for this topic (you may skip this section on move on the question section)
We wanted to start a group containing members from 5th standard to masters graduates and working professionals. Also a few business people. The group size is limited to 25 to begin with.

The idea is to see how different members understand and respond to a questions. It would depend on existing knowledge, own strengths, method of visualization etc

This simple topic happened to become a great experience.

The question:
How do we interpret or visualize a fraction. for example 3/5. I had problem with fractions for long..

Simple answer and visualization:
- rewrite the same as 3 * 1/5.
- reread the fraction as "divide 1 piece into 5 pieces and take 3 of them out of it".

Perspective of an artist:

Take one piece of dough, divide it into 5 balls (balls are not shown of equal size but the idea is captured) and roll 3 of them into a chappathi or poori.

Perspective of a doodler and a youngster

An youngster just getting introduced to fractions.

Got it right in dividing one piece into 5. And drawn with love to their parents. superb.

Though this representation is not a correct mathematical answer, it laid foundation for new learning.

Seeing 1/5 as dividing a circle into 5 equal parts. She just learned about degrees and circles. only children can break the monotony. She took a part of the question.. Just the 1/5 part.

Fraction Multiplication:

Actually, application of multiplication of fractions was not in the scope of this question. But once we got the cake drawing from rithu, we have got a better understanding of fractions.

Another example:

Take 3 apples, cut it into 5 equal parts and share it with 3 people.

There is nothing wrong with this answer. Mathematically correct. But, a bit difficult to visualize.

Though the topic looks trivial, visualizing fraction was a major problem in my childhood.

The world of fractions becomes interesting with irrationals (square root of 2, 3 etc) and transcendental's (pi and e). Let us get to know them in the following articles.

Idea to Commercialization 2016 - Scheme 4

Idea to Commercialization

A unique opportunity for final year students of Coimbatore district

Selected individuals and teams shall receive support for mentoring and funding through our venture building partners.

If you think your project work had one of these elements

·  Innovative (technical or business)

·  Cost effective

·  Alternate process

·  Had a research element

Who can apply?

Education:           Only current Final year students.

Stream:                Agriculture, Arts, Commerce, Diploma, Engineering, Law, Medicine, Science

Region:               Candidates representing institutions from Coimbatore District

·                                                                  Masters and PhD Candidates can also apply

Last date for registration: 4 May 2016

Shortlisted applicants will be invited for a presentation within 31^st May 2016.

Make It Happen - Scheme 3

An initiative to augment the team for our science and research institute

Still open: last date for registration 30 April 2016

Water From Air - Scheme 2

An initiative launched in April 2016 to identify teams along with mentoring and funding support.

This is still open: Last date for registration is 01 May 2016.

Play with gravitational constant - scheme 1

This was a first of our outreach initiatives announced in September 2015, to identify talent and augment our team.

As a bootstrap and in early days, we got a very good reach but only one response (may be we did not ask the right question). And interestingly, the person who responded is part of our team now.

However, it ended up as an article in this blog - Field of Gravitation

Euclid makes another point. Symbolification.

Article status:                              Draft
Time Estimate for Reading:        15 min
Learning Objectives:                   Euclid's Division Lemma, Class X, NCERT, Mathematics
Effort Required:                          low
Pedagogy Model:                        Symbolification of mathematics
Prior Math Tools:                       Secondary school level Arithmetic (long division)

It happened to me. i always thought or was told, mathematics deals with numbers. Open any mathematics book, there are less and less of numbers and more and more of symbols.  brain sees it, but mind does not accept it. a mental block. easiest way is to runaway from mathematics.

I kept running away from mathematics until Isaac Asimov (The greatest science fiction author) and Prof Shankar Ramamurthy (Yale) stopped me. Isaac Asimov introduced me to the concept of formula-analysis (we have used this in our articles on science and discovery) and Prof Shankar Ramamurthy introduced me to the power of symbolification. The 'ification' may be new to you. we like it that way. just that it sounds and rhymes good. There are more of this in our Tamil language.

What better way to start than asking, "why x is the unknown?".

Invest a little of your time. 3 min and 57 sec.  watch this TED video by Terry Moore.

Having got to terms with x, let us move ahead and try to deal with more symbols. Let us begin with Euclid again. The Euclid's Division Lemma.

Euclid's division Lemma.
Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that
a = bq + r
and
0 ≤ r < |b|,
where |b| denotes the absolute value of b

Feel like running away (i used to. but not now). please resist your temptation. it is not that difficult. At a first sight, it may look incomprehensible, just give it a try. There is nothing wrong in giving it a try.

All that Euclid has done is symbolification of division.

take a few examples of division.
5/5; the quotient (q) is 1, remainder (r) is 0,5 the numerator (n) and 5 the denominator (d).
11/5; the quotient is 2, remainder is 1.
4/5;   the quotient is 0 and remainder is 4
10/5; the quotient (q) is 2, remainder (r) is 0,10 the numerator (n) and 5 the denominator (d).

We chose these examples such that we had the 3 scenarios. equal to, less than, greater than. we will be using this in most of physics.
n = d     | r =0
n > d     | r ≠ 0
n < d     | r ≠ 0

Observations:
- we find that remainder can never be greater than denominator.
- take the example 11/5.  11 (n) =  5(d) * 2 (q) + 1(r)

Now let us retain these symbols and take away the numbers

therefore, n = d*q + r; where 0 <= r <= |d|

That's Euclid's division Lemma, a = bq+r.   just replace a with n and b  with d.

The question is why do we take so much pain? why cant we leave it as such?

There are quite some advantages in symbolification.
1. It is like a rule that is followed by all integers.
2.  The Euclidean division lemma is extended into an algorithm to find out the Greatest Common Denominator. It is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 147 = 252 − 105.

The range of applications are more. please explore the reference section.

Welcome to the world of symbolification. you may want to call it algebra. it makes life easier.

Note:
We will be adding a few solved examples and exercise problems from NCERT text book later.

References:
https://en.wikipedia.org/wiki/Euclidean_division
https://en.wikipedia.org/wiki/Euclidean_algorithm
https://en.wikipedia.org/wiki/Euler%27s_totient_function

understanding the euclidean algorithm
https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/the-euclidean-algorithm

https://www.khanacademy.org/computing/computer-science/cryptography/modern-crypt/v/the-fundamental-theorem-of-arithmetic-1

Exercise
https://www.khanacademy.org/quetzalcoatl/deprecated-exercises/deprecated-bucket-5/v/the-fundamental-theorem-of-arithmetic

Method of generating primes
https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/sieve-of-eratosthenes-prime-adventure-part-4


Primes density vs natural logarithm
https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/prime-number-theorem-the-density-of-primes



                                     A critical review of textbooks by Mr Badri Seshadri

Do exams kill Creativity?

Please post your comments for this post or mail your **responses to aatrral@gmail.com



Note:
I have to honor my words. I have decided (that's a decent word instead of  ' forced to') not write any more articles on education or entrepreneurship in my previous article drop-out and drop-in. There may be others in competition like educational institutions, parents, teachers, industry etc. You may refer to this article for some resources.

** The space below is intentionally left empty. to compile your responses. Please.

drop-out and drop-in

In many ways, this article is important for me as a individual. may be for people who happened to read my previous articles as well. the torture comes to an end. not for me but for my friends. Friends who encouraged me to write. 

I feel very, very happy. though i admit and understand that happiness or sorrow is just a state of mind, i just want to allow emotions to take over.

Why should i end writing? don't know. but there was a "message", an "intuition", people use different words. But the writing was clear on the wall. Just one last article and henceforth write only about science, science and science. nothing else.

My usual method is to write about something. in the whole of the article, one thing or the other seems to stand out and that becomes the title. But this time, as it was the last of the article on education, philosophy and entrepreneurship, i had to just connect the dots. so started this article with a title.

As i was writing this article, a few events unfolded and would wish to present it as it happened. The event itself happened to be the core of the idea that i was trying to communicate.

There was a call. it was from one of my good friend.

Am leaving the content at the end of this article. unedited. it is incomplete in every sense, but it is ok. some people say, do it right the first time. i keep trying. but somehow, it happens to be incremental. write, rub, write, reorg... until convinced

Here is the conversation with my friend.

Friend: i am planning to start a health and fitness center. 

My mind: I know you have been working on it for the past 10 years. you are a living example. have reduced from 115 kgs to 75 kgs. cycling as a hobby. regular workouts and food freak. i could still recall inviting him once to our home in bangalore. he happened to travel all the way from Coimbatore, carrying a food basket. On the breakfast table, he opened it. it felt like....... (you may fill in the blanks)

Friend: During these years i had many unanswered questions. in fact a thousand of them. Was confused. But recently i seem to have got to the core. had done some intensive research and this idea seems to fit. But still, i would like to validate with a few people who wish to give an audience.

Me: I would like to attend your program. when is it?

My mind: I am not bothered if he is a doctor, his educational background, what is the certificate that he carries? He has lived it, experienced it. what better teacher or mentor can i ask for.

Me: I too had been struggling like you to understand science. To fit kinetic energy into the scheme of things it took 4 years. and it took another 2 years to understand that acceleration is core of the idea. once we begin from there, whole of science becomes comprehensible. the reason for newton apple story becoming so famous. we happen to remember the story and leave the concept alone.

Friend: You have been into teaching? how should i structure my session?

My Mind: What is there to prepare? you are just going to share your experience. you can answer most of the questions that are raised. with examples. spontaneously. 

Me: First 10 min is important. How much time people are gonna sit? what is the objective? what is your experience with health and fitness. The remain part of the time will evolve into a unique experience.

The same topic on another day, different audience, different questions. does not matter. It will sure be as interesting as the first session. lively. 

I can't wait to attend his program. we got to wait for good things. it seems to happens that most of the time, . i am looking forward to have a great experience.  From a master.

There is nothing outside my boy (girl). Everything is within us. It is no more inside-out. it is outside-in. a master leads you 'in'. rather drops you inside of you. drop-in.

what a way to end? no. no. just drop-out and drop-in.

In case you are interested to know what i was writing before i got the most important call

------------------------------------------------------------------------------------------------------------------

We have heard many stories of dropouts. The most famous being Steve Jobs and Bill Gates. All stories point to one thing. They dropped out of education system and dropped into; rather immersed themselves into learning. That's definitely learning for life.

I have not taken examples from my hometown. most of us are drop outs. Mentally. Not physically. 

We respect our society. we respect our parents. Children seem to be more magnanimous. They sacrifice and compromise for the sake of elders.

People think we don't have courage. not really. pick any child at random, listen and we will not only hear it. we will feel it. I have. "Appa likes it", "Amma said", "The child next door does it", "A cousin does it",... 

Moral stories may be the cause. May be because they are taught only in schools, it is applicable only for school students. It is not in syllabus for elders.

Just take a look at the road. choa's. i dont want to talk about corruption and get myself into trouble. it does not exist. i am afraid. 

But we want every child to observe strict discipline, be perfect, excel in every field, nothing less than a 100....

The situation in higher studies is not much different. just about crossing teens. recognized as majors by age. another 3, 4 or 5 years of strict obedience. most of the energy is spent in sitting, for most part of the day. Sitting is most tiring activity for youngsters. body and brain does better only when it is used. May be because we live in a mechanized world, we want it sparingly used. We drain all the energy and push them to perform. Ironic.

With that said, the major reasons for drop out.

- Every one is unique. this leads to interest mismatch and not lack of interest.

- The quotient variation. Intelligence, Emotional, Creativity and Memory. Personalization is the key.

- Assessment driven learning. it should be the other way. Self-assessment.

- Attendance and monitoring. Kills self-responsibility and self-esteem.

May be we have a solution. at least for "majors" moving into higher education. for those interested in learning. we have access to best of the teachers from across the world. 

All you need is an internet connection and a some kind of device with which you can view and program. You may have the device on a desk, on your lap or even on your head (head mounted devices). Virtual and Augmented reality are already available. its only a question of affordability. Start with wikipedia or you tube. Depends on whether you like text or visual

That's all fine. What about the "certificate". ok. Maybe register with a distance learning program. We will require them for syllabus, exams, practicals and projects. With proliferation of internet, distance learning is not distant anymore. Rather it is closer. 

Days are not far away. Learn from different resources. for example, khan academy tracks the complete history. they are issuing badges. how many exercises you have solved, how many chapters you have completed, what is the topic that you last learned. 

Looks like we just send this link and people can make out our interest levels, duration of learning, approach, almost everything. including your behavior. They are mining data. 

Even IIT has certificate courses. Once we have collected sufficient credits, some organization may even provide a degree certificate. 

Schools: NOS and IGCSE.

Arts and humanities: 

Sciences:

Engineering:  AMIE

Medicine: Alternative medicine 

Law: 

My favorite picks for science.

Schools:    www.khanacademy.org   (started electrical and computer engineering)

Higher Ed:  NPTEL, coursera, edx, yale, MIT, princeton, Amrita, hyperphysics, wikipedia, you tube links, homibabha, vigyan prasar, ncert

------------------------------------------------------------------------------------------------------------------------

Cutoff Vacation. Try Hackation.

You may read the title as cutoff vacation or "cutoff vacation". The first one is much better than the second.

Cutoff. The word that gives nightmares for children. It definitely sends shivers through the spine. We are referring to cutoff in the context of marks. 

Thank god, i lived a generation earlier. The vacations we had are still fresh in our memory. At least we longed and lived for the days. The days of no rules. no plans. The only plan was "not to have a plan". Though short lived, it had the inertia to see through the grueling, forthcoming year.

The cutoff works in many ways. it's something like a Swiss military knife. Has the ability to cutoff many interesting things from the life a child. It also cuts many things off - parents, industry and society. It seems to cutoff the relationship with every one and everything. Even the very subjects they are supposed to learn.

The typical itinerary for "cutoff vacation" -.

- Complete the next year syllabus in this vacation (what do we do for the whole year).

- Put them into a coaching class so that they can catch up  (Brain is not a dust bin to dump things)

- Do something useful (what are we doing the whole year)

So, this time don't give your child the "cutoff vacation". Try Hackation.

A typical itinerary for Hackation

- begin with enjoying laziness. Just do things very, very slowly. do nothing at all. 

- catch up with friends, relatives, movies... there are many more. what ever the child feels is important

- Give responsibility (no attendance and timelines please). 

- Allow them to explore . See the child as to have a mind of its own. we have got used to see "it" as a brain. Effect of computers and its processors.

Let children decide what they want to do. let them make mistakes. allow them to correct themselves. remember we (parents, teachers, society) have been controlling, instructing, pushing, pestering and 'inging' them on many other things; for most part of the year. The children are expected to live our life. Lets understand, they have a life of their own. Scientists talk about universe or multi-universe. Every child has its own world or its own universe. May be beyond that.  we don't know. 

Just ask them about their world.  It would definitely be very rich. We will be surprised. Pleasantly.

Given time, space and responsibility, (just a few that i came across) 

An unbelievable social transformation

                                     

An attempt to bring power to rural villages

                                      

Self-Responsibility

So, lets give them the break. they seem know more than us. it is their generation. just keep faith in our children. given a choice, they will settle only for the best. The best in the sense of their inherent strength or potential. 

In the age of technology, with information at the finger tips, global teachers.. it is a decision to make. Elders (grown physically) have to make a decision. 

The decision is to allow children to make their own.

Let them enjoy "hackation". A life long vacation.

How to Hack Creativity - Interview with Self

Some much with so less. Just 15 Secs Video

One of my friends sent me this link and this made me think. rather i should say wonder.

But why did this make such a deep impact on me?  It has to do with what i am doing. let us just go ahead with this; on a lighter note.

Just a hypothetical interview with myself. When no one is there to ask questions, better ask ourselves. A hack from "Six thinking hats" of Edward Debono

1. Question: What is the nature of your business?
Answer:   Education, Research and Entrepreneurship
Critic:      Well, you don't seem to have focus.
My View: I am not able to distinguish between all three. They are inseparable. did my best to explain it in "The story of a tagline".
Hacked from: From personal experience.

2. Question: Whom do you think is your customer?
Answer:    Any one interested in learning. From school, college or industry
Critic:       Well. You don't seem to have a target customer
Answer:    If my view 1 is valid, this should be valid.
Hacked From: Euclid's Elements,

3.  Question: What is your administrative policy?
Answer:  A happiness centered organization
Critic:     Well. The whole world is talking about Professionalism, Smartness and Profitability.
My view:   A Happiness centered environment would be more creative.
Hacked from: 13 year old logan. Please spare 11 min and 13 secs. In fact the title was hacked from this kids use of the word "Hack Schooling".

                         
4. Question: What is the road map for your institution?
Answer:  Establish an evolution based science park (Education), Create an infrastructure for tele- robotics (Research), Support for innovators (Accelerator)
Critic: You need a big tag and big money.
My View: There is always a way out. Take on a participative approach.
Hacked From: Rather than hacked, i would prefer to use the word inspiration. There are many inspiring stories of groups going on to create better homes, companies, villages, cities and even countries.

5. Question: How do you categorize your institution?
Answer: Social Entrepreneur. There are different "preneurs". Entrepreneur, Edupreneur, Techpreneur etc. I even ended up creating few more. The Studentpreneur and Teacherpreneur
Critic:    That is a tough game. Either be a for profit or a non-profit. This is midway.
My view: Money is required. But it is not the only requirement.
Hacked From:  Quote "Don't Sell. Assist people to Buy"

6. Question (based on question 3): What is your opinion on creativity?
Answer: The strategies are Think like an Inventor (Re-invent the wheel), Imagine as though the problem or solution is obsolete (make it obsolete) and more importantly just keep working with an open mind.
Critic: If things are not broken, don't try to fix it.
My View: Creativity comes from our experience and openness. So the core idea is to redefine the problem in the way we understand.
Hacked From: Creativity Crisis


7. Question: How did you come up with this title?

If you have come this far, you would really appreciate the difficulty i go through in communicating to the world of what i am or my company is doing. It has been a long journey.

Why go that distance? for that matter, i had difficulty in introducing myself.  All that i could do was come up with an article "The hard part of life is to introduce oneself"

The question that i raised to myself was, How to be creative?. That eventually led me to ask the opposite question of what is not creativity (using our making it obsolete principle)?

So, this is where it took me. There were articles on the role of educational institutions, parents, teachers and organizations affect creativity.

How educational institutions kill creativity - close to 400 million views.

https://www.ted.com/talks/ken_robinson_says_schools_kill_creativity?language=en

How Teachers kill creativity

https://www.goshen.edu/art/ed/creativitykillers.html

How parents kill creativity

http://www.metroparent.com/daily/parenting/parenting-issues-tips/whats-killing-creativity-in-kids/

http://www.idiva.com/news-iparenting/11-things-parents-do-that-can-crush-their-kids-creativity/15081822

http://playfullearning.net/2014/06/kill-childs-creativity/

Whether it is parents or schools that kill creativity  ( books shown in the video are great ones)

https://www.youtube.com/watch?v=IZWWqYVGanI

How companies kill creativity

http://www.onlinemba.com/blog/13-ways-companies-kill-creativity/

With that said. what is the lesson learnt?
I have considered unschool, deschool, alternate school, home school. Now, i have landed into "hack school". But a school in the current sense deals mostly with teaching and does not have much emphasis on research and entrepreneurship.

May be we should apply the principles of re-invent and make-it-obsolete.

homeschool, homeoffice..... what about home college? looks like it is leading somewhere...

Name of the business entity seems ok. Aatrral represents energy. The initial tag line was "Identify Potential. Create Value". After about 2 years, the tag line has evolved into "The temple of acceleration". More on this here ....  ("The story of Tagline"). As we evolve, we might change again.

May be we have not asked the right questions as yet.

The search continues........ I have to come up with something like the 15 sec clip (a hack again) that we began with.

Foot Note:
came up with this article as an after thought.
http://openneuron.blogspot.in/2016/04/cutoff-vacation-try-hackation.html

Joule has different faces

Article status:                              Draft
Time Estimate for Reading:        15 min
Learning Objectives:                   Concept of Moment and Work
Effort Required:                          Medium
Pedagogy Model:                        Evolution based
Prior Physics Concepts:             displacement, velocity, acceleration, mass, power, force
Prior Math Tools:                       Secondary school level Arithmetic, geometry and algebra


It has been a great journey so far. Starting from space, time and mass, we have come quite some distance from tower of Pisa and Galileo. or should i say displacement?

Looking back at our journey,

Considering only space and time, we added displacement, velocity and acceleration to our vocabulary. That is all an object can do. May be remain at rest or continue in motion. Motion being of uniform velocity or of uniform acceleration. We carefully avoided 'jerk' in our journey. jerk is non-uniform acceleration and is not a comfortable experience.

Different objects, distinguished by its mass can be in motion. So, motion should take the mass of object into consideration. so, we discussed about power of motion and power of rest. The power of rest introduced us to the concept of force. The concept of force led us to the concept of mass and with that we were able to distinguish between mass, weight and force.

With that in perspective, there are few more concepts that we may add to our vocabulary. As much as we deduced force from power of rest, we shall also deduce the the concept of Work and Energy from the power of motion.

We shall look at the concept of power objectively  and also take a peek at the way the concept of work evolved. The concept of energy has to wait for one more article.

Objective perspective:
power = mass * acceleration * velocity

Power is measured in Watt's.

When there is more than one variable (m, a and v in this case) mathematicians, scientists and physicists always like to play with them or try different combinations. shall we say cook things with the same ingredient but different recipes.

p = m * a * d/s   (velocity is m/s)

What are the different combinations. Mathematically speaking we have two choices. Use permutations or combinations. We choose to use combinations instead of permutations (as order is not important).

mass * a has been given the word force .  Force measured in Newton's.

- mass/second does not seem to make any meaning (in fluids we shall introduce mass flow rate).
- acceleration per second or rather change is acceleration per second has been carefully ignored
- mass * displacement also does not seem to make any meaning.

Can we make out something by F * d (force into displacement). A big Yes. But not from manipulating the formula. It comes from analyzing the concept of Lever. Lever would be the first of simple machines. The famous Archimedes (300 BC) statement of "Give me a Lever and the earth shall be lifted"

                                              


Examining the picture of Archimedes with lever, The triangle, rather a 3 dimensional prism, is the support point (fulcrum). The effort is to be applied by the hand of Archimedes. The load to be lifted is the earth.

For a little review of the historical importance, you may take a look at this link.

Now is the time to introduce the concept of rigid body. In the articles so far, we have been discussing about a particle at rest or in motion. A particle necessarily we assumed as a spherical object, with all of its mass concentrated at a single point.

A rigid body need not be spherical. It can have different shapes. May be imagine balancing a broom stick
                        

Observe that the support point is not the geometrical center, but shifted to one side. The point on the object at which support is provided is called the center of mass or center of gravity (since gravitational acceleration is constant at the surface of earth).

Request you to take some time off to watch some videos on application of this lever and center of mass principle. The Equipois and Steadycam. Salute to Mr Garett Brown, the self taught innovator.

The principle of the lever:

                                  
For the lever to be equilibrium (equilibrium is the word used for rigid body at rest),
- provide a support at the center of mass.
- The product of weight and distance on either side should be equal.

This product of W and d is given a word called moment.  We may replace Weight with Force. Force is a more generic word and provides a more generic definition for moment. Moment is the product of force and perpendicular distance from the point of support.

At this juncture, recall that, there is another word called momentum which is a product of mass and velocity. Watch out the effect of adding 2 letters 'um'. It leads to a totally different meaning.

We were discussing on the concept of rest when we considered a particle. Similarly, for a rigid body to be at rest we need to introduce the concept of equilibrium.  So, for a rigid body to be at rest, it should be in both force equilibrium (center of gravity should not move) and moment equilibrium (rigid body should not rotate).

Having discussed on the concept of center of mass, moment, force equilibrium and moment equilibrium let us move forward to get to the concept of work.

Using the concept of moment equilibrium (otherwise called as lever principle), we can
lift any object. may be that is why Archimedes said he can lift the earth. Just put in the mass of earth (5.972 × 10^24 kg) and mass of Archimedes (approx 100 kg) in the formula. We would require a lever of length approx 5.972 × 10^22 meters. Compare this with circumference of earth 4.0075 * 10^4 m. we need a lever roughly 20 orders of magnitude.

Evolution of Work as a concept in physics:

Leaving that aside, we also understand from our experience that, nature does not offer any free lunch. To gain something we need to loose something. 

                               

Observe the displacement of load and displacement of effort. Force * displacement along the line of application of force. Effort has to applied over a longer distance to lift the load through a shorter distance.  So, it is only natural for physicist's to come up with another concept. The concept of Work. Work is the product of Force and displacement along the line of application of force

W = F * d. Force in Newton's and displacement in meters. The unit of work is Nm. Like force is measured in Newton's, Work (Nm) is measured in Joules. Work is just an abstract concept but as we move on, we shall understand the importance of it application in modeling physical systems.

With the introduction of work, Power gets a proper definition. it is the rate of doing work.

P = W/s and is measured in watt's.

For the sake of completeness, Simon Stevin's inclined plane experiment seems to have helped establish the concept of Work. With this experiment, Stevin used the concept of Weight * displacement along the inclined plane. Earlier to this people seem to have tried to establish a relationship with force and angle of inclination.

We began with Galileo (Momentum=mv), Watt (Power = w/s), Newton (Force=ma) and now we have reached Joule (Work=Fd).

But Joule seems to have multiple faces. To know about the different faces of Joule, we need to understand yet another abstract concept called energy. Rather potential and kinetic energy. 

Joule's different Faces:

Moment                  = f*d   measured in Nm

Work                       = f*d  measured in Nm  again

Potential Energy     = m*g*h measured in Nm  again

Kinetic Energy        = 1/2 m*v^2  measured in Nm  again

We have been considering only straight line motion so far. once we complete our discussion on energy (potential and kinetic) in the following article, we can connect the dots and clear the missing links.

By now we have collected enough tools to tread into the field of 2 dimensional rotational motion. There are 2 more concepts which have the same unit as Nm.

Torque                     = Force * radius measured in Nm

Couple                    = Force *distance measured in Nm again

There are 6 concepts in physics with the same unit of Nm. Joule seems to have six faces and it would be nice to know them all. On a lighter note, just imagined joule with six faces and wondered how joule would listen and talk to six people at a time. Really enjoyed the thought experiment. Leave alone the challenge of we talking to a person with six faces.

Now back to physics, this was the area which took a lot of time to crack. it took close to 7 years for me to get to terms. it was a tough journey. may be my IQ levels are low. End of the day, happy that i made it this far.

Let us move on with our journey into yet another abstract concept; Energy. 

References:
https://www.math.nyu.edu/~crorres/Archimedes/Lever/LeverLaw.html

Mass, Weight and Force

Article status:                              Draft
Time Estimate for Reading:       20 min
Learning Objectives:                  Difference between mass, weight and force
Effort Required:                         Medium
Pedagogy Model:                       Evolution, Formula Analysis
Prior Physics Concepts:             displacement, velocity, acceleration, mass, power, force
Prior Math Tools:                       Secondary school level Arithmetic, geometry and algebra

Space, Time and Mass. The three fundamental concepts on which the whole of science is built. rather that is how we try to understand nature.

We have discussed on space in the article "Euclid has Point". We are yet to discuss on time. The focus of this article is mass.

Mass, Weight and Force. This a cause of confusion for most beginners of science. And there have been many efforts from different perspectives to give a clear understanding of this. This is one such effort from yet another perspective (after all each one learns from a different perspective).

By definition, weight is perceived gravity and force is a general term referring to the product of mass * acceleration. mass has multiple definitions and are summarized at the end of this article.

Similar to mass and weight, for beginners in electricity, the direction of conventional current, electromotive force (which is not a force. we should not even say work. we should say it is potential difference) etc are some pitfalls to watch out for.

This is one of the simplest examples i could come across to understand the difference between mass and weight. or rather to define weight.

By now, you should have got it. if not, continue reading.

Let us begin with mass. The textbook definition says "amount of matter". What is the matter with that definition?

From our experience, if someone asks us to buy, say "salt", we will ask how many grams or kilograms. depending on the shop, assuming we were living in the days of Galileo, it may be measured with a simple balance or a spring balance. The digital scales are yet to arrive.

The word 'mass' is yet to have a clear definition. The term heaviness and weight were used for comparing quantities. So, we could say the weight or heaviness of salt is 1 kg.

We are also aware of the fact from Galileo's experiments that every object on the surface of earth has the same acceleration; independent of their heaviness.

Newton with his experiments and observation finds that gravitational acceleration is different on earth and moon. If we carry the same quantity of salt to moon and weigh it with spring balance, it will show a different value as the gravitational acceleration of moon is different.

This is not an acceptable situation. Say, for example we weigh gold with a spring balance and carry it to a place which is much higher in altitude, it would show a different value. With increase in altitude, the gravitational acceleration would decrease. you may refer to the graph on this page for variation of gravitational acceleration with altitude. Notice here that the simple balance is a comparison device and can be used to measure weight independent of altitude.

This issue was put to rest by newton. He introduced the concept of force which is equal to mass * acceleration. f= ma.

At this point, for a detailed discussion on f=ma, you may refer to the article http://openneuron.blogspot.in/2016/03/acceleration-bridge-between-space-time.html

On earth, f = m * 9.81 m/s^2.

What if we define the unit of force as 1 Newton which is equal to 1 kg mass * 1 m/s^2. That is 1 Newton of Force.

So, going back to our salt example which we measured with spring balance, it will contain the acceleration component of 9.81 m/s^2.

Try converting this weight into Newtons. We have to multiply 1kg by 9.81m/s^2
which is = 9.81 Newtons.

Now we can say that mass of the object = 1 kg and Weight (or force) of the object is 9.81 Newtons.

This brings us to another question. Is weight and force the same?

No. Weight has been given an new definition. Weight is defined as perceived gravity. What does that mean?

This is where we take help from E M rogers toy. Alternative example is of measuring our weight in a lift.

When the toy is at rest, the weight of the ball (mg) and force exerted by the spring (kx), balance each other. But, when we allow the toy to fall down, the force exerted by the spring will remain the same, but weight of the ball will reduce and the ball will be pulled by the spring to the top. The component of 'g' will have a lesser value and hence the weight becomes lesser. So, we can say that weight of an object changes with accelerated (or decelerated) motion and hence the definition "perceived gravity ".

We have discussed about weight and understood the concept of perceived gravity with the simple toy.

Now, discussing about force, force is a general term. it may be applied to gravitation force, electric force, magnetic force etc. Weight is similar to force but used only in the context of gravitational force.

Having settled the matter with weight, force and mass, the story does not end here.

Almost all practical applications use weight for calculations. So, what is the need for mass? Every object we take for analysis has a gravitational force acting on it which forces us to take ‘weight’ for analysis. So, is mass just a theory?

Imagine in next 100 years, we colonize few galaxies (am I being too optimistic?). You weigh something here and transport them to some planet in Andromeda. Obviously, the gravitational acceleration there would be different which depends on the mass of the planet (Newtons law of gravitation G*(m1*m2) / r^2). That’s a different story. You export gold and they decide not to pay what you ask for, since it weighs way less than in earth. Chaos!

This would force us a generalized term for measurement. Generalization is possible only if we could remove the variable. In this case, acceleration due to gravity. Remove that ‘g’ from weight and we are left with mass.

In order to make life more interesting, Mass has different names or definitions.
mass:                       density/volume  (we would discuss this in fluids)
Gravitational mass: m * g
Inertial Mass:          m * a  (linear motion):  m=f/a.
Inertial Mass:          mr^2 * alpha  (rotational motion). Here inertial mass is mr^2 and not just m.
Amount of matter:  Given a mass m, we can find the number of atoms or molecules present.                                                 Avogadro number is the link.

                               Mass = moles * Atomic Mass Unit  (1 mole of a substance will weigh AMU gms)
                               As much as 1 dozen is 12, 1 mole represents a number which is 6.023 * 10^23

                               So from our example, we may ask the question, how many molecules of NaCl are                                present in 1 kg mass of salt.

                               From the periodic table, we find that the molecular mass of  NaCl is                                                     22.989 + 35.453 grams.
                              These many grams will contain 1 mole or 6.023 * 10^23. From this we can find                                   out the amount of matter or number of molecules contained in 1 kg mass of salt.
Relativistic Mass:  The change is mass at higher velocities










The story of mass has is yet to see its end.  It seems to make a beginning. Einstein comes in and says that mass and energy are convertible and related by the most famous equation E = mc^2. The Mass energy equivalence.