pwiki - User contributions [en]

Newton's Law of Universal Gravitation

2013-10-11T12:37:22Z

Redmonds: /* URLs */

''Kreshnik Angoni''

* The law of Universal Gravitation is formulated by Newton for two point particles with masses <math>m_1</math> and <math>m_2</math> at a distance r between them. The magnitude of force exerted on one particle by the other one is given by: 

[[image: Kreshnik_N_Univ_Grav_Fig1.png|right]]

<math>F_{12} = F_{21} = G\frac{m_1m_2}{r^2}</math> 

The measurements show that the universal constant of gravitation is <math>G = 6.67*10^{-11}Nm^2/kg^2\,\!</math> 

* The gravitational force is a vector directed versus the source that exerts this force. So, its vector form is: 

<math>F_{12} = - G\frac{m_1m_2}{r^2}\frac{\vec{r_{21}}}{r_{21}}</math> 

where <math>\frac{\vec{r_{21}}}{r_{21}}</math> (or, <math>\hat{r}_{21}</math>) is the unit vector with tail at mass <math>m_2</math> – the source of <math>\vec{F}_{12}</math>. 

<math>F_{21} = - G\frac{m_1m_2}{r^2}\frac{\vec{r_{12}}}{r_{12}}</math> 

<math>\frac{\vec{r_{12}}}{r_{12}}</math> (or, <math>\hat{r}_{12}</math>) is the unit vector with tail at mass <math>m_1</math> – the source of <math>\vec{F}_{21}</math>. 

* To apply the gravitational law for two bodies close to each other one must use integration techniques and
the difficulty of calculi depends on the form of the two bodies. But, if the bodied are far enough to each
other, one may model them as point particles and apply the law in its original form. 

* In particular, with some approximations, we are able to model the interaction of earth with an object on its surface as if the whole mass of the earth is concentrated at its center and the objects are at distance <math>r =R_{earth}</math>. 

* Many experiments have shown that; when several particles interact gravitationally between them the
'''''principle of linear superposition applies'''''. So, inside a system of particles <math>m_1,m_2,m_3</math>,…<math>m_n</math> , the force exerted on mass <math>m_1</math> is: 

<math>\vec{F_1} = \sum_{i = 2}^n \vec{F}_{1i}</math> 

===THE GRAVITATIONAL AND THE INERTIAL MASS===

*When expressing the second law of Newton we use the inertial mass: 

<math>\vec{F}_{NET} = m_{in}\vec{a}</math> 

When formulating the gravitation law, Newton was not sure that the mass of particles in this law is the
same as their inertial mass. Let’s verify this issue. 

We start by supposing that the mass in the gravitational law may be different from <math>m_{in}</math>. So, we call it <math>m_{gr}</math>. 

Let’s consider now a body in free fall close to earth surface. The earth will exert on it the gravitational force with magnitude: 

<math>F_{gr} = G\frac{m_{gr}M_{Earth}}{R_{Earth}^2}</math> 

Here we assume that the body is close to the surface so that its distance form earth center is Earth ≅ <math>R_{Earth}</math> 

As this is the net force exerted on the body we apply the second law of Newton 

<math>F_{NET} = m_{in}a = F_{gr} = F_{gr} = G\frac{m_{gr}M_{Earth}}{R_{Earth}^2}</math> 

So, we get 

<math>m_{in}a = m_{gr}(\frac{GM_{Earth}}{R_{Earth}^2}) = m_{gr}g</math>, where <math>\frac{GM_{Earth}}{R_{Earth}^2} = g</math> 

Then, 

<math>a = \frac{m_{gr}}{m_{in}}g</math> 

A big number of measurements show that the acceleration of free bodies is equal to <math>g = 9.8m/s^2</math>. This
means that <math>m_{gr}/m_{in} =1</math>, and <math>m_{in} = m_{gr}</math>. 

So, the experiments confirm that the gravitational mass is the same as the inertial mass. 

* Let us apply the gravitation law for the force exerted by earth over a mass 1kg close to earth. 

<math>\vec{F} = - G\frac{M_{Earth}}{(R_{Earth}+h)^2}*\frac{\vec{R_{Earth}}}{R_{Earth}} \equiv \vec{g}</math> 

So, the g-vector is equal to the gravitation force exerted on a mass 1kg. 

By measuring the force exerted on the mass 1kg in different locations on the earth one gets a whole system of g-vectors (fig.2). The totality of these vectors forms the gravitational field of the earth. 

[[image: Kreshnik_N_Univ_Grav_Fig2.png|left]]

 Note that the g-vector magnitude decreases with the increase of distance “h” from the earth but it is always directed versus the center of the earth. The gravitational field of the earth has a spherical symmetry. 

In fact, it is not exactly spherical, because the model of earth as a uniform density sphere is not very precise. Now, the object weight is equal to gravitational force exerted by this field 

<math>\vec{W} = m\vec{g}</math> 

So, the weight of the same object is a vector that is different in different points of gravitational field of the earth. 
 

===KEPLER’S LAWS ON PLANETARY MOTION===
 

* '''First law''': The planets move on elliptic orbits around the sun that is located at one of its focuses. 

[[image: Kreshnik_N_Univ_Grav_Fig3.png|left]]

The minor axis is long 2b and the major axis is long 2a. The closest distance to sun is called perihelion and the biggest distance to sun is called aphelion (fig.3). 

*'''Second law:''' The line sun- planet sweeps out equal areas for equal interval of times. 

[[image: Kreshnik_N_Univ_Grav_Fig4.png|TOP]] 

*'''Third law:''' The square of the period of planet motion is proportional to the cube of average distance from the sun. Calculations show that the average of distance sun-planet is equal to half of major axis a. Then, satellite motion tells that 

<math>T^2 = k a^3</math> where 

<math>k_{sun} = \frac{4\pi^2}{GM_{sun}}</math> 

'''Note :''' Kepler’s laws are valid for elliptical paths of any planet around a central body; for example the
moon moving around the earth but in this case 

<math>k_{Earth} = \frac{4\pi^2}{GM_{Earth}}</math> 
 

===THE ENERGY OF PLANETS===
 

[[image: Kreshnik_N_Univ_Grav_Fig5.png|left]]

* As the mass of other planets is much smaller than the mass of sun
we neglect their action on the motion of the studied planet. We
consider that the system sun-planet is a conservative system, i.e. the
forces originated from outside it are zero. 

In these circumstances: 

a) The torque of exterior force is zero and we can apply the principle
for conservation of angular momentum. 

b) The work done by Net exterior force is zero and we can apply the
principle of energy conservation for the system '''sun-planet'''. 
 

* The principle of angular momentum conservation tells that <math>\vec{L_A} = \vec{L_P}</math> 

or <math>\vec{r_A} \times \vec{p_A} = \vec{r_P} \times \vec{p_P}</math> 

The equality of magnitudes brings to condition 

<math>r_Ap_ASin90^o = r_Pp_PSin90^o \Rightarrow r_Am_{planet}v_A = r_Pm_{planet}v_P</math> 

So, we get 

<math>r_Av_A = r_Pv_P\,\!</math> 

* The principle of energy conservation tells that that <math>E_A = E_P\,\!</math>

As the mechanical energy is '''E = K + U''' where 

<math>K = \frac{m_{pl}v_{pl}^2}{2}</math> and 

<math>U = -G\frac{m_{pl}M_{Sun}}{r_{pl-sun}}</math> 

we get 

<math>\frac{m_{pl}v_{A}^2}{2} - G\frac{m_{pl}M_{Sun}}{r_{A}} = \frac{m_{pl}v_{P}^2}{2} - G\frac{m_{pl}M_{Sun}}{r_{P}}\,\!</math> 
This gives, on cancelling <math>m_{pl}</math> from the equation: 

<math>\frac{v_{A}^2}{2} - G\frac{M_{Sun}}{r_{A}} = \frac{v_{P}^2}{2} - G\frac{M_{Sun}}{r_{P}}\,\!</math> 
which leads to: 

<math>2GM_{Sun}(\frac{1}{r_P}-\frac{1}{r_A}) = v_{P}^2 - v_{A}^2</math> 

Now, we know <math>r_Av_A = r_Pv_P\,\!</math> 
and the fact that: 
<math>r_A + r_P = 2a\,\!</math> 

Thus, 

<math>v_A^2 = \frac{GM}{a}\frac{r_P}{r_A}</math> and, 

<math>v_P^2 = \frac{GM}{a}\frac{r_A}{r_P}</math> 

Finally, by substituting one of this expressions at the expression of total energy (at perihelion or aphelion)
we get 

<math>E = -G\frac{m_{pl}M_{Sun}}{2a}</math> 

''See also'' 

*[[Dynamics of Circular Motion]] 



===EXERCISES===
*[[Exercises on Satellites and Binding Energy]] 

===Online Resources===
* [http://www.greenfudge.org/2011/11/28/video-incredible-inuit-hunt-mussels-under-sea-ice/ Video: Incredible Inuit hunt mussels under sea ice]

Choosing the Right Career Path

2012-10-08T14:06:32Z

Redmonds:

About 2 years ago, one of my friends went through a career crisis and asked a bunch of us for help. This page started as a list of book summaries in an e-mail to my friend. In a way, my career search will never truly be over though, so I have added a few things and tweaked a few others over the ears.

- ''Scott Redmond, 8-Oct-2012''

==Book: What Should I Do With the Rest of My Life, by Po Bronson==
This book contains more than 60 stories of people who have switched careers and found something that's a better fit. There's a consistent theme that it takes time and effort to identify what you should do with your life. I don't mean to say that it's overwhelming; more that it's worthy of sustained effort and it's okay to try things and potentially make mistakes. This book was also very helpful for showing how normal it is to struggle to find something better, simply by sharing so many stories. It also highlighted that very few people learn their true gift through an epiphany; almost everybody starts out with a small voice inside that they must carefully nurture before it turns into a driving force.

==Book: Do What You Love==
This book was written by a guy who went from a $200k+ job on wall street to a $50k income doing a mix of his favourite types of work. It focuses on working on what you love to do and offers a lot of advice about the transition, with stories of other people who have made similar transitions and love what they do. There are some sections on dealing with fear of transition, and the key idea that I still remember is to feel the fear and do it anyway. I've since learned other techniques for dealing with fear (see The Work of Byron Katie below).

== The Work of Byron Katie ==
Byron Katie helps people find truth.

Her approach is deceptively simple (just 4 questions and a turnaround) but powerful for dealing with fears, anxieties, and any stressful thoughts or beliefs.

There's a great overview at [http://www.thework.com/index.php thework.com] and a blog (with videos) at [http://www.byronkatie.com/ www.byronkatie.com]. I would recommend starting with "Loving What Is" or "I Need Your Love, Is That True?" because both books include a great introduction to the process. I've tried doing the approach for 30 days in a row and it really does have a lasting effect. I've let fear control my decisions in the past; this gives me the upper hard.

==Podcast (and book): Get-It-Done Guy==
Stever Robbins, the Get it Done Guy (host of a weekly podcast on iTunes - yes, I'm a huge productivity geek) paraphrased Martha Beck's approach (below) as doing more of the things that feel like eating oreo ice cream cakes and less of the things that feel like eating worms. Check it out: [http://getitdone.quickanddirtytips.com/finding-life-purpose.aspx Get-It-Done Guy: How to Choose Your Life Purpose].

When I first started listening to Stever Robbins, I found his style annoying but the information was great so I kept listening. After a few months, I was hooked. The podcast gives a new tip every week for free and you can search through all of the previous tips. I highly recommend his book as well (although you can get much of its content from the podcast archives if you're willing to search a bit). Two of my favourite things from his book are: 1) creating a life map to help maintain balance, and 2) evaluating your technology.

==Book: Steering by Starlight, by Martha Beck==
Although I'm a big fan of the 7 Habits of Highly Effective People, I always struggled with the 2nd step of identifying a personal mission. Beck's approach is a bit more touchy-feely but it helped me much more because it's such a different perspective.

== Working with a Career Advisor ==
Working with a career advisor helped a lot too. I've already mentioned that mine put a lot of weight on the MBTI, and one of the biggest benefits for me was finally accepting that it's okay to make big decisions based on feelings instead of wasting time trying to develop a formula to "prove" that it's the right choice. The real point is that it was helpful in ways I hadn't expected, so I would recommend talking to a career advisor if you're at all interested. One of the problems I had with all the books was that as I got more and more information, some of it conflicting, the problem seemed to expand. My career advisor didn't address this directly but I did come out of it with a much clearer understanding.

==Book: Great Work, Great Career, by Stephen Covey==
This book makes a strong argument for identifying your key strengths and behaving like a consultant (in terms of providing solutions -- not in terms of 90h work weeks) which is a theme that came up in many different books. I especially liked their suggested layout for breaking down your strengths, into Talents, Passions, Conscience (and I also listed weaknesses - since you don't necessarily have to make them better but you do have to manage them). Disclaimer: I'm still reading and studying Covey's 7 Habits of Highly Effective People so I'm definitely biased in his favour.

==Book: Do What you Are==
This book uses the MBTI personality types (not to be confused with the Kiersey types that use the same letters) and offers suggestions based on your type. This approach was heavily recommended by my career advisor, and she was emphatic that your type just identifies your preference so you're/we're not limited to the careers suggested for any one type. It's similar to being right- or left-handed: you can open a door with either hand, but it's more comfortable one way. The advice for each type was most helpful when I evaluated what I thought was useful and threw out the rest.

== My Own Wandering Career Path ==
When I was choosing a university program, I wanted to enter computer engineering because I liked playing video games and I wanted to be a computer programmer. My cousin suggested a different program, systems design engineering, because I could still do computer engineering if I wanted but it was much more flexible so I could also transition into electrical or mechanical engineering if I changed my mind. The flexibility was good, because I enjoyed the mechanical engineering courses more than the others, and by my 3rd year my dream job was to program computer simulations for robots. After undergrad, I did a master's degree (also in systems design engineering) focused on simulating robotics - and that's when I discovered my true gift: I worked as a teaching assistant to help pay my student loans and I found myself developing new teaching material and giving extra tutorials for ''fun''. I was hooked on teaching, but I wanted to have some working experience before becoming a teacher. After my masters degree, I worked as a mission operations analyst for MDA Space Missions, simulating the robotic arms on the International Space Station. It was my 3rd year dream job, but after the initial excitement of working with NASA and learning lots of new things wore off, the desire to teach returned. After 5 years with MDA, I became a physics teacher in continuing education at Vanier. The teaching was great fun and very intense with longer hours even than when we had space shuttle missions going on. A year later, facing the arrival of a second child, I went back to engineering at MDA with its family-friendly working hours (typically 7:00 to 3:30). Over the next year, I took some time off to research career changes and work with a career counselor. I was really lucky to find some fantastic resources that clarified what I was doing and gave me tools to work through it. I ultimately returned to Vanier to see how things work out, and don't regret it for a second.

When I was in undergrad, one of my favourite professors told us all that lifelong jobs and single-path careers no longer existed; that our generation would change careers as often as our parents' generation changed jobs (within the same career). As an undergrad I found that scary. Having lived through it, it can be daunting and even frightening but once you work through the fear it becomes empowering and exciting!

I'll finish with one of my favourite quotations, which I encountered during this research: 
'''''Work is love made visible''''' 
—Kahlil Gibran

Things I wish I knew before my first job

2012-10-08T12:26:22Z

Redmonds:

''Scott Redmond''

This is a somewhat random collection of things that I wish I'd known before I started my first job. 

== Selecting your career based on your true gifts ==
I went to school for engineering because I happened to be good at math and science, and I liked working with technology (i.e. playing video games). It wasn't until I was a TA (teaching assistant) during my masters program that I realized my true gift was actually helping people learn.

Many career descriptions sound cool and interesting, but lasting satisfaction comes from exercising your true gift.

"Work is love made visible." Kahlil Gibran

Identifying ''your'' true gift seems to be the hardest step for most people. One approach is to think of something you could do for hours without getting bored. Is it something that others may find difficult or frustrating? If so, that's a clue that it's one of your gifts. Another approach, from my favourite productivity podcast ([http://getitdone.quickanddirtytips.com/finding-life-purpose.aspx Get-It-Done Guy: How to Find Your Life Purpose]), is roughly to do more of what feels like eating oreo ice cream cake and less of what feels like eating worms. It's basically the same approach recommended by other life coaches, but these visuals help you tap into the part of your conscience that knows what you really want but is often very quiet.

If you can't think of what your true gift might be, focus instead on trying things that you're interested in. As you experience a wider variety of what life can offer, you'll get closer and closer to finding out what it is that you love to do. The important thing is to keep moving; getting stuck in a rut is what leads to boredom.

For more tips, see [[Choosing the Right Career Path]].

== Get-It-Done Guy ==
When I first started listening to Stever Robbins, I found his style annoying but the information was great so I kept listening. After a few months, I was hooked. The podcast gives a new tip every week for free and you can search through all of the previous tips.

Here are some of my favourites:
* [http://getitdone.quickanddirtytips.com/finding-life-purpose.aspx Get-It-Done Guy: How to Choose Your Life Purpose]
* [http://getitdone.quickanddirtytips.com/how-to-memorize-quickly.aspx Get-It-Done Guy: How to Memorize Quickly]
* [http://getitdone.quickanddirtytips.com/how-to-study-back-to-school.aspx Get-It-Done Guy: How to Study as You Head Back to School]

== TED Talks ==
We're very lucky to have access to amazingly successful people right at our fingertips. These videos stand out.
* [http://www.ted.com/talks/richard_st_john_s_8_secrets_of_success.html Richard St. John's 8 secrets of success]
*: This is a short but inspiring video laying out the key elements for success.
* [http://www.ted.com/talks/richard_st_john_success_is_a_continuous_journey.html Richard St. John: "Success is a continuous journey"]
*: Another one that's short and sweet.

== Autofilter in Excel ==
This one feature makes Excel super helpful for working with large sets of data (e.g. for lab results). I often have tables with 10,000 rows and 100 columns, and the Autofilter makes it easy to look at a single subset at a time. For example, to answer a question like, "What was the biggest force observed on the payload during the last 60 seconds of all simulations where the canadarm was braked?"

The Autofilter feature also makes it easy to use Excel for project checklists. If each row is a task, you can easily filter out all tasks that are already done (which is better than deleting them because you retain the information about complete tasks) or look at specific subsets (e.g. all tasks that are assigned to Bob and due in the next 3 weeks).

There are other more powerful ways to do this (e.g. relational databases) but this is easy to learn and Excel is easy to find. Even if you don't have Excel, the Autofilter feature exists in nearly all spreadsheet programs (e.g. OpenOffice).

== Versioned Backups ==
Like CVS, subversion, DropBox, or Time Machine, etc... The point is to have some tool that keeps track of changes you make over time. Even DropBox can do this so it doesn't have to be complex to set up. Someday you'll accidentally save over a good file or your hard drive will crash and then you'll need the time history of backups that you've set up.

This is most important just before a deadline, when it would really hurt to have to retype what you've already done.

Try to make it automatic so it won't take your energy away from the actual document. DropBox is awesome for this, and it's free.

== Wiki as a collaboration tool ==
Especially personal ones like tiddlywiki for organizing information, and online wikis that allow group collaboration. This is fantastic for group projects, where one of the biggest challenges is how to transfer knowledge from experienced group members to newbies.

One of my favourite parts of university was working on student projects (e.g. solar car, F1 racer, aerial robotics, etc...), and one of our biggest problems was sharing information within a large group. Wikis didn't exist back then, but they would have been great!

== Outliner and/or Mind Mapping Software ==
Outliner and mind mapping software lets you group things and lay out ideas to whatever level of detail you want. Then you can collapse and expand to see the high-level view.

One of the best uses for this is when you have a bunch of information and don't know how to organize it. You can just lay out each idea, fact or quotation as a separate item, and reorganize the whole set into categories however you like.

There are a bunch of free apps for this, for Windows, Mac, iOS, etc... Just google "outliner" or "mind mapping" for more information. These are also easy to with pen and paper!

What I find funniest about this is that, although the outlining and mind mapping tools are ''so'' simple, people with decades of experience are often amazed when they see the results and how easy it was to get those results.

== Edward Tufte ==
One day our counterparts at NASA suddenly had simple, effective, graphics for their analysis - Tufte's work was what made them so much better. Tufte's book called ''The Visual Display of Quantitative Information'' has been especially helpful. A few simple tips for preparing graphics (like the ratio of data in the graphic to the amount of ink used to print it) have powerful effects in the final product. This sort if thing is very subtle and most engineers don't even think about it so it can give you an edge.