The IIT-JEE reforms sought to be introduced by the human resources development ministry have sparked a national debate in India.
While the HRD ministry states the new exam pattern is an attempt to curb students' over-dependence on coaching classes and to bring some respectability to the board exams, coaching classes and a section of students question the need for introducing the reforms in the middle of the academic year without prior warning.
This year, about 198,000 students appeared for 4,935 seats at the seven Institutes of Technology; the Institute of Technology-Banaras Hindu University; and the Indian School of Mines, Dhanbad.
In the wake of these reforms, Professor V G Idichandy of IIT-Madras discusses with Yogesh K Upadhyaya, the reasons for a change in the IIT-Joint Entrance Examination pattern and the eligibility criteria.
Prof V G Idichandy, convener of STF was the chairman of JEE, IIT-Madras in 1999. He is currently a professor in the department of Ocean Engineering at IIT-Madras and the dean of students.
In February 2004, the directors of IITs decided to set up a special task force to recommend reforms for the JEE. Prof Idichandy was the convener of the Special Task Force. Being JEE chairmen at different times, all the STF members were aware of the issues that needed to be sorted out.
They collected a lot of relevant information, had consultations and submitted a report to the government. The present reforms for JEE were deliberated upon at different levels and they represent the collective efforts of all IITs. Excerpts from the interview:
Could you explain the changes proposed in IIT-JEE pattern and eligibility criteria for appearing in the exam?
The decisions approved by the ministry of human resource development are:
• JEE will continue to be the sole criterion for ranking the students for admission to IITs, IT-BHU, ISM Dhanbad and other participating institutes.
• School results will be factored into the admission process for the IITs. Only those students who secure a first class or equivalent in the plus-2 examinations will be eligible for admission to the IITs.
• For students belonging to SC/ST there will be a relaxation of 5 per cent of marks below the level of marks prescribed in the respective examination.
• JEE will be a single objective type examination. The question paper can include short write-up on a topic in physics, chemistry or even mathematics followed by objective type questions that are based on the write-up.
• A student can write the JEE in the year in which he/she passes the XII standard examination and/or in the following year.
• Those who join any of the IITs, IT-BHU, and ISM Dhanbad through JEE-2006 will not be permitted to appear for JEE in the future.
• The procedure will be reviewed after JEE-2006. A one-time exception would be allowed for those who passed their qualifying examination in 2005 or earlier.
What prompted the expert panel to change the pattern of exam, from two stages (screening and main) to only one-step exam?
The committee which suggested the screening test ('98 and '99, JEE committees) recommended a review of the examination system after five years. The screening test is a test of the aptitude of the student while the main examination is a test of his/her analytical ability.
We have completed six JEEs after this system was introduced. The IIT system is convinced that an objective type examination can be designed to test both aptitude and analytical ability. Based on this the special task force recommended a comprehensive objective test over a two stage examination.
We must keep in mind that clearing JEE is not an end in itself but it is meant for the very limited purpose of selection of students to an academic programme.
There is a restriction on appearing for more than two times in the exam. There is also a restriction on appearing again, if someone has secured a seat at any of the seven IITs, IT-BHU, and ISM. Why?
The main reason for adopting these two measures is to provide a level playing field for candidates appearing for JEE. Now the JEE stipulates a first class in higher secondary school as the minimum eligibility requirement.
Are we justified in allowing candidates with additional years of preparation to take the exam along with fresh students just out of the school? The candidates appearing two or three years after leaving the school get additional time for JEE-specific preparations, have an altogether different mindset and approach to life and education, when compared to those directly out from the schools.
Two or three years of intensive preparation in specific subjects does not make these candidates any wiser as the training is more to beat the examination rather than for the pleasure of learning.
More or less the same reason is valid for those who have already joined the IITs. In addition, they deny opportunity for others who would have gained entry into one of the IITs.
There were protests in New Delhi and other cities. Why didn't they protest when the main exam-type pattern was changed to two-stage exam (main plus screening) exam in the year 2000?
In the last six years, the IIT system has received much more visibility and recognition. Many students and parents believe that an IIT degree is a passport to success and fame. Those who prepare the students for JEE at astronomical costs, exploit this attitude or misconception to their advantage and would like a predictable, easily recognizable pattern for JEE, and are scared of changes.
I don't believe that the stage-managed protests in some cities were done by students. It does not require much research to identify those behind the protesters. Just like any other unjustified protests, agitation against changes in JEE has been instigated by people with vested interests.
The students and their parents should realise the fact that there are equally good/better opportunities outside the IIT system.
How will a two-tier system help students?
The two-tier JEE tests the aptitude of students for professional education through a screening test and their analytical ability through the main examination.
Many a time the parents decide what their children should study. We have come across students who get admission because of rigorous coaching. Later, they struggle to complete the programme as they have no aptitude for an engineering education. We feel that one way of countering this situation is by allotting branches of study at the end of second or third semester and not at the time of admission as practiced today.
Such a procedure will give the students enough time to study various options available to them and choose the branch most suited to their aptitude. I hope the IITs, in due course, will look at this suggestion seriously.
On the requirement of minimum 60 per cent marks in the XII board exam, critics argue that different boards give marks in different ways. For instance, CBSE is considered lenient in grading, while it is relatively difficult to achieve first class in some boards, such as Uttar Pradesh and Bihar. Also, one of the high school boards in eastern India immediately announced the plan to grade the XII standard exam liberally, so that no one from the state is denied admission to IITs. How will you counter this?
Ideally, normalised scores of the different boards should be used. With about 40 school boards in our country, collecting results, arriving at normalised scores, and above all, convincing the stakeholders of different eligibility marks for different boards are all Herculean tasks -- next to impossible, in fact -- considering the time factor involved.
First class (60 per cent) in a qualifying exam is much more simple, easily understood and accepted.
Though the STF recommended normalised scores, the Standing Committee of the IIT Council (SCIC) which includes the directors of the IITs as members decided on first class in higher secondary as minimum criteria and we saw a lot more wisdom in that approach. Our aim was to give adequate importance to school education in our country. We are extremely happy that all the school boards whole-heartedly welcomed the move.
Don't you think the timing for the announcement of reforms was wrong, as many students have already enrolled in coaching classes, and they are in the middle of an academic year in high school?
IITs cannot introduce reforms in JEE for the convenience of those in the coaching classes. The invariant pattern of the entrance examination of the premier institutions made possible the mushrooming of coaching institutions. They grew because of the demand for quality higher education, seats for which are too few in our country.
Now it has reached a stage where genuine students with raw intelligence and aptitude armed with the knowledge gained in schools find it difficult to compete and gain admission to the premier institutions.
The main target is the student presently in the XII standard or equivalent ready to take an examination, irrespective of the pattern and with all its surprises.
Every year JEE is announced in the month of September. This year was no exception. Eligibility requirements and the type of examination are always explained in the information brochure included along with the application form.
It is reported that over 95 per cent of the students who clear IIT-JEE, have joined coaching classes. This is an unusually high number compared to other national-level exams, such as AIEEE. Why there is a rush for this exam? How will the new reforms help curb the coaching class dependence for an IIT-JEE aspirant?
I have already explained the reasons for the rush for JEE and the consequent enrolment in coaching institutions. We are convinced that the present reforms in JEE will reduce the necessity for additional preparation.
IITs should be allowed to conduct JEE 2006 with the present reforms to prove the point. If the panic reactions witnessed in some cities where coaching centres are concentrated are some indications, the IITs have already won the first round!
Please tell us more about IIT-JEE; when it was introduced and why it has an objective format.
JEE has a long history. It started in the early sixties and evolved through years. There were changes of various sorts over the years. For instance, English was a compulsory paper once upon a time. Later, English, though compulsory, was not in the reckoning for deciding the ranks.
Questions in Hindi and an option for answering in Hindi and regional languages were introduced during eighties. Objective type questions were a part of the JEE for some time. Screening tests also went through metamorphoses before the present pattern was adopted in the year 2000.
JEE is well known for its credibility and the IITs consider JEE very sacred and are very proud of its autonomy.
IITs take extra care and precautions to keep the credibility intact and the faith of the general public and students in the JEE system notwithstanding the incident in 1997 that shook the very foundations of JEE.
But with the strong support of the ministry of HRD, IITs outlived the threat and restored all that was lost within a short time. The whole country recognises the independence of JEE and importance of IITs and that is precisely why anything about IITs receives so much public attention. This public audit makes us more responsible and to perform better.
Don't you agree that school education has faced a setback with students concentrating more on how to crack the entrance tests?
Yes, school education has become a big casualty, as there is no credit for twelve years of schooling for admission to the institutions of higher education, be it state-run or central. (Tamil Nadu is perhaps the only state, where equal importance was given to both school performance and entrance test for admission to professional courses.
Though the TN government announced admissions to professional courses based on school results from 2005, because of the procedural issues, it was not implemented).
Why should anyone take school education seriously, when admissions to higher education is based on one entrance test and the eligibility for admission to national institutions is a just a pass in qualifying examination?
Under such a situation, it is but natural for a student to spent time preparing for entrance examination rather than in the school.
What changes do you foresee once these reforms are implemented?
The measures recommended to reform JEE will pave the way for visible changes in the attitudes of students to the school education. When JEE questions are of the level of XII standard, students and parents will realise the futility of going through the rigours of coaching institutions and neglecting school education.
We also feel the rural students and girls will benefit from these reforms but only the future would tell how far we are justified in making these assumptions. Girls constitute only 8 per cent of the student population in IITs (IIT-Madras taken as reference). But school results show better performers are girls, so this could be the beginning of a new trend.
Tuesday, April 1, 2008
Problem Solving in Physics
Introduction
The problems and examinations in this physics course exercise not only your knowledge of physics but also your skill in solving problems. Professional physicists earn their salaries not particularly for their knowledge of physics but for their ability to solve workplace problems. This document presents tips for honing your problem solving skills. These tips and techniques will prove useful to you in your physics courses, in your other college courses, in your career, and in your everyday life.
To set the stage, I want to discuss an example of problem solving from everyday life, namely building a jigsaw puzzle. There are a number of different approaches to building a jigsaw puzzle: My approach is to first turn all the pieces face up, then put together the edge pieces to make a frame, then sort the remaining pieces into piles corresponding to small "sub-puzzles" (blue pieces over here, red pieces over there). I build the sub-puzzles, then piece the sub-puzzles together to build the whole thing. Other people have different approaches to building jigsaw puzzles, but nobody, nobody, builds a puzzle by picking up the first piece and putting it in exactly the correct position, then picking up the second piece and putting it in exactly the correct position, and so forth. Solving a jigsaw puzzle involves an approach--a strategy--and a lot of "creative fumbling" as well.
Your physics textbook contains many solved "sample problems". The solutions presented there are analogous to the completed jigsaw puzzle, with every piece in its proper position. No one solves a physics problem by simply writing down the correct equations and the correct reasoning with the correct connections the first time through, just as no one builds a jigsaw puzzle by putting every piece in its correct position the first time through. The "solved problems" in your book are extraordinarily valuable and they deserve your careful study, but they represent the end product of a problem solving session and they rarely show the process involved in reaching that end product. This document aims to expose you to the process.
Solving a physics problem usually breaks down into three stages:
Design a strategy.
Execute that strategy.
Check the resulting answer.
This document treats each of these three elements in turn, and concludes with a summary.
Strategy Design
Look before you leap. Whenever you face a problem, there is an immediate temptation to rush in, roll up your sleeves, and begin tinkering with it. Resist that temptation. If you start your detailed work--the execution stage--immediately, you will likely write down a lot of correct statements that do not lead to an answer. Instead, think about the problem on an overview level. What sort of conceptual tools will you need to solve the problem? What path will you take to the solution, and in what direction should you start off? Concretely, it often helps to classify your problem by its method of solution.
If you are looking for a child lost in the woods, your first step is to sit down, think about what the child probably did and where he probably is, and devise a strategy that will allow you to effectively rescue him. If, instead, you just rush about the woods in random directions, you're likely to become lost yourself.
Where are you now, and where do you want to go? Before you can design a path that takes you from the statement of the problem to its answer, you must be clear about what the situation is and what the goals are. It often helps to check off each given datum of the problem, and to underline the objective. But for getting an overall sense of the problem, nothing beats summarizing the whole situation with a diagram. The diagram will organize your work and suggest ways to proceed. One of my course graders told me that "When students draw a diagram and label it carefully, they are forced to think about what's going on, and they usually do well. If they just try a globule of math, they mess up."
Keep the goal in sight. Don't get caught in blind alleys that lead nowhere, or even in broad boulevards that lead somewhere but not to where you want to go. It sometimes helps to map a strategy backwards, by saying: "I want to find the answer Z. If I knew Y I could find Z. If I knew X I could find Y . . . " and so forth until you get back to something you are given in the problem statement.
Some students find it useful to make a list of the information given and the goal to be uncovered (e.g. "given the constant acceleration, the initial velocity, and the time, find the displacement"). Others find it sufficient to write down only the goal (e.g. "to find: displacement").
Ineffective strategy. Do not page through your book looking for a magic formula that will give you the answer. Physics teachers do not assign problems in order to torture innocent young minds . . . they assign problems in order to force you into active, intimate involvement with the concepts and tools of physics. Rarely is such involvement provided by plugging numbers into a single equation, hence rarely will you be assigned a problem that yields to this attack. In those rare instances when you do face a problem that can be solved by plugging numbers into a formula, the most effective way to find that formula is by thinking about the physical principles involved, not by flipping through the pages in your book.
Make the problem more specific. You're asked to find the number of ways that M balls can be placed into N buckets. Suppose you can't even begin to map out a strategy. Then try the problem of 3 balls in 5 buckets. Solving the more specific problem will give you clues on how to solve the more general problem. And once you use those clues to solve the more general problem, you can check your solution by trying it out for the already-solved special case M=3 and N=5.
Large problems. At times you will be faced with big problems for which no method of solution is immediately apparent. In this case, break your problem into several smaller subproblems, each of which is simple enough that you know how to solve it. At this strategy-design stage it is not important that you actually solve the subproblems, but rather that you know you can solve them. You might begin by mapping out a strategy that leads nowhere, but then you haven't wasted time by implementing this strategy. Once you have mapped out a strategy that leads from the given information to the answer, you can then go back and execute the calculations. This strategy has been known from the time of the ancients under the name of "divide and conquer".
Execution (Tactics)
Eventually, of course, you do have to roll up your sleeves and tinker with the problem. As you do so, keep your strategy in mind, and keep the following tips in mind as well:
Work with symbols. Depending on the problem statement, the final answer might be a formula or a number. In either case, however, it's usually easier to work the problem with symbols and plug in numbers, if requested, only at the very end. There are three reasons for this: First, it's easier to perform algebraic manipulations on a symbol like "m" than on a value like "2.59 kg". Second, it often happens that intermediate quantities cancel out in the final result. Most important, expressing the result as an equation enables you to examine and understand it (see the section on "Answer Checking") in a way that a number alone does not permit.
(Working with symbols instead of numbers can lead to confusion as to which symbols represent given information and which represent unknown desired answers. You can resolve this difficulty by remembering--as recommended above--to "keep the goal in sight".)
Define symbols with mnemonic names. If a problem involves a helium atom colliding with a gold atom, then define mh as the mass of the helium atom and mg as the mass of the gold atom. If you instead pick the symbols m1 and m2, you stand a good chance of mixing up the symbols and their meanings as you solve the problem. And if you don't define the symbols at all, but just begin throwing around m's and M's, you'll confuse both yourself and whoever is grading your answer.
Keep packets of related variables together. In acceleration problems, the quantity (1/2)at2 comes up over and over again. This collection of variables has a simple physical interpretation, transparent dimensions, and a convenient memorable form. In short, it is easy to work with as a packet. Take advantage of this ease. Don't artificially divide this packet into pieces, or write it in an unfamiliar form like t2a/2. Packets like this come up in all aspects of physics--some are even given names (e.g. "the Bohr radius" in atomic physics). Look for these packets, think about what they are telling you, and respect their integrity.
Neatness and organization. I am not your mother, and I will not tell you how to organize either your dorm room or your problem solutions. But I can tell you that it is easier to work from neat, well-organized pages than from scribbles. I can also warn you about certain handwriting pitfalls: Distinguish carefully between t and +, between l and 1, and between Z and 2. (I write a t with a hook at the bottom, an l in script lettering, and a Z with a cross bar. You can form your own conventions.) These suggestions on neatness, organization, and handwriting do not arise from prudishness--they are practical suggestions that help avoid algebraic errors, and they are for your benefit, not mine. (On the other hand, it doesn't hurt to be neat and organized for the benefit of your grader. One course grader of mine pointed out: "If I can't read it, I can't give you credit.")
Avoid needless conversions. If the problem gives you one length in meters and another in inches, then it's probably best to convert all lengths to meters. But if all the lengths are in inches, then there's no need to convert everything to meters--your answer should be in inches. In fact, you might not actually need to convert. For example, perhaps two lengths are given in inches and the final answer turns out to depend only on the ratio of those two lengths. In that case, the ratio is the same whether the lengths going into the ratio are inches or meters. It's easy to make arithmetic errors while doing conversions. If you don't convert, then you don't make those errors!
Keep it simple. I will not assign baroque problems that require tortuous explanations and pages of algebra. If you find yourself working in such a way, then you're on the wrong path. The cure is to stop, go back to the beginning, and start over with a new strategy. (Generations of students have kept track of this rule by remembering to KISS: Keep It Simple and Straightforward.)
Answer Checking
Checking your answer does not mean comparing it to the answer in the back of the book. It means finding the characteristics of your answer and comparing them to the characteristics that you expect. Some of your problems--particularly the ones assigned early in the course--will actually lead you through the checking stage in order to familiarize you with the process. Other problems will leave it to you to perform this check. In either case, checking your answer is not just good problem solving practice that helps you gain points on problem assignments and on exams. The checking stage builds familiarity with the content of physics and the character of problem solutions, and hence develops your intuition to make solving other problems--and learning more physics--easier. (See Daniel F. Styer, "Guest comment: Getting there is half the fun", American Journal of Physics 64 (1998) 105-106.)
Dimensional analysis. Suppose you find a formula for distance (in, say, meters) in terms of some information about velocity (meters/second), acceleration (meters/second2), and time (seconds). If your formula is correct then all of the dimensions on the right hand side must cancel so as to end up with "meters".
Numerical reasonableness. If your problem asks you to find the mass of a squirrel, do you find a mass of 1,970 kilograms? Even worse, do you find a mass of -1,970 kilograms?
[Reasonable speeds. "My calculations give me a speed of 23 m/s. Is this reasonable?" It's hard for most people to get a feel for the reasonableness of speeds expressed in meters per second. Until this qualitative feel develops, Americans should check for reasonableness by converting speeds in meters per second to speeds in miles per hour: simply double the number (20 m/s is about 40 mi/hr). Non-Americans should convert to kilometers per hour: simply quadruple the number (20 m/s is about 80 km/hr).]
Algebraically possible. Would evaluating your formula ever lead you to divide by zero or take the square root of negative number?
Functionally reasonable. Does your answer depend on the given quantities in a reasonable way? For example, you might be asked how far a projectile travels after it is launched at a given speed with a given angle. Common sense says that if the initial speed is increased (keeping the angle constant) then the distance traveled will increase. Does your formula agree with common sense?
Limiting values and special cases. In the projectile travel distance problem mentioned above, the range is obviously zero for a vertical launch. Does your formula give this result? If you solve a problem regarding two objects, does it give the proper result when the two objects have equal masses? When one of them has zero mass (i.e. does not exist)?
Symmetry. Problems often have geometrical symmetry from which you can determine the direction of a vector but not its magnitude. More often they have a "permutation" symmetry: If your problem has two objects, you can call the cube "object number 1" and the sphere "object number 2" but your final answer had better not depend upon how you numbered your objects. (That is, it should give the same answer if every "1" is changed to a "2" and vice versa.)
Specify units. "The distance is 5.72" is not an answer. Is that 5.72 miles, 5.72 meters, or 5.72 inches? Similarly, if the answer is a vector, both magnitude and direction must be specified. (The direction may be drawn into a diagram rather than stated explicitly.)
Significant figures. Any number that comes from an experiment comes with some uncertainty. Most of the numbers in this course come with three significant figures. If a ball rolls 3.24 meters in 2.41 seconds, then report its speed as 1.34 m/s, not 1.34439834 m/s. Most introductory physics courses do not require a formal or technical error analysis, but you should avoid inaccurate statements like the second quotient above.
Large problems. If you break up your large problem into several subproblems, as recommended above, then check your results at the end of each subproblem. If your answer to the second subproblem passes its checks, but your answer to the third subproblem fails its checks, then your execution error almost certainly falls within the third subproblem. Knowing its general location, you can quickly go back and correct the error, so its effects will not propagate on to the remaining subproblems. This can be a real time-saver.
Summary
The problems in your physics course can be fun and exciting. Approach them in the spirit of exploration and they will not disappoint you!
Strategy design
Classify the problem by its method of solution.
Summarize the situation with a diagram.
Keep the goal in sight (perhaps by writing it down).
Execution tactics
Work with symbols.
Keep packets of related variables together.
Be neat and organized.
Keep it simple.
Answer checking
Dimensionally consistent?
Numerically reasonable (including sign)?
Algebraically possible? (Example: no imaginary or infinite answers.)
Functionally reasonable? (Example: greater range with greater initial speed.)
Check special cases and symmetry.
Report numbers with units specified and with reasonable significant figures
The problems and examinations in this physics course exercise not only your knowledge of physics but also your skill in solving problems. Professional physicists earn their salaries not particularly for their knowledge of physics but for their ability to solve workplace problems. This document presents tips for honing your problem solving skills. These tips and techniques will prove useful to you in your physics courses, in your other college courses, in your career, and in your everyday life.
To set the stage, I want to discuss an example of problem solving from everyday life, namely building a jigsaw puzzle. There are a number of different approaches to building a jigsaw puzzle: My approach is to first turn all the pieces face up, then put together the edge pieces to make a frame, then sort the remaining pieces into piles corresponding to small "sub-puzzles" (blue pieces over here, red pieces over there). I build the sub-puzzles, then piece the sub-puzzles together to build the whole thing. Other people have different approaches to building jigsaw puzzles, but nobody, nobody, builds a puzzle by picking up the first piece and putting it in exactly the correct position, then picking up the second piece and putting it in exactly the correct position, and so forth. Solving a jigsaw puzzle involves an approach--a strategy--and a lot of "creative fumbling" as well.
Your physics textbook contains many solved "sample problems". The solutions presented there are analogous to the completed jigsaw puzzle, with every piece in its proper position. No one solves a physics problem by simply writing down the correct equations and the correct reasoning with the correct connections the first time through, just as no one builds a jigsaw puzzle by putting every piece in its correct position the first time through. The "solved problems" in your book are extraordinarily valuable and they deserve your careful study, but they represent the end product of a problem solving session and they rarely show the process involved in reaching that end product. This document aims to expose you to the process.
Solving a physics problem usually breaks down into three stages:
Design a strategy.
Execute that strategy.
Check the resulting answer.
This document treats each of these three elements in turn, and concludes with a summary.
Strategy Design
Look before you leap. Whenever you face a problem, there is an immediate temptation to rush in, roll up your sleeves, and begin tinkering with it. Resist that temptation. If you start your detailed work--the execution stage--immediately, you will likely write down a lot of correct statements that do not lead to an answer. Instead, think about the problem on an overview level. What sort of conceptual tools will you need to solve the problem? What path will you take to the solution, and in what direction should you start off? Concretely, it often helps to classify your problem by its method of solution.
If you are looking for a child lost in the woods, your first step is to sit down, think about what the child probably did and where he probably is, and devise a strategy that will allow you to effectively rescue him. If, instead, you just rush about the woods in random directions, you're likely to become lost yourself.
Where are you now, and where do you want to go? Before you can design a path that takes you from the statement of the problem to its answer, you must be clear about what the situation is and what the goals are. It often helps to check off each given datum of the problem, and to underline the objective. But for getting an overall sense of the problem, nothing beats summarizing the whole situation with a diagram. The diagram will organize your work and suggest ways to proceed. One of my course graders told me that "When students draw a diagram and label it carefully, they are forced to think about what's going on, and they usually do well. If they just try a globule of math, they mess up."
Keep the goal in sight. Don't get caught in blind alleys that lead nowhere, or even in broad boulevards that lead somewhere but not to where you want to go. It sometimes helps to map a strategy backwards, by saying: "I want to find the answer Z. If I knew Y I could find Z. If I knew X I could find Y . . . " and so forth until you get back to something you are given in the problem statement.
Some students find it useful to make a list of the information given and the goal to be uncovered (e.g. "given the constant acceleration, the initial velocity, and the time, find the displacement"). Others find it sufficient to write down only the goal (e.g. "to find: displacement").
Ineffective strategy. Do not page through your book looking for a magic formula that will give you the answer. Physics teachers do not assign problems in order to torture innocent young minds . . . they assign problems in order to force you into active, intimate involvement with the concepts and tools of physics. Rarely is such involvement provided by plugging numbers into a single equation, hence rarely will you be assigned a problem that yields to this attack. In those rare instances when you do face a problem that can be solved by plugging numbers into a formula, the most effective way to find that formula is by thinking about the physical principles involved, not by flipping through the pages in your book.
Make the problem more specific. You're asked to find the number of ways that M balls can be placed into N buckets. Suppose you can't even begin to map out a strategy. Then try the problem of 3 balls in 5 buckets. Solving the more specific problem will give you clues on how to solve the more general problem. And once you use those clues to solve the more general problem, you can check your solution by trying it out for the already-solved special case M=3 and N=5.
Large problems. At times you will be faced with big problems for which no method of solution is immediately apparent. In this case, break your problem into several smaller subproblems, each of which is simple enough that you know how to solve it. At this strategy-design stage it is not important that you actually solve the subproblems, but rather that you know you can solve them. You might begin by mapping out a strategy that leads nowhere, but then you haven't wasted time by implementing this strategy. Once you have mapped out a strategy that leads from the given information to the answer, you can then go back and execute the calculations. This strategy has been known from the time of the ancients under the name of "divide and conquer".
Execution (Tactics)
Eventually, of course, you do have to roll up your sleeves and tinker with the problem. As you do so, keep your strategy in mind, and keep the following tips in mind as well:
Work with symbols. Depending on the problem statement, the final answer might be a formula or a number. In either case, however, it's usually easier to work the problem with symbols and plug in numbers, if requested, only at the very end. There are three reasons for this: First, it's easier to perform algebraic manipulations on a symbol like "m" than on a value like "2.59 kg". Second, it often happens that intermediate quantities cancel out in the final result. Most important, expressing the result as an equation enables you to examine and understand it (see the section on "Answer Checking") in a way that a number alone does not permit.
(Working with symbols instead of numbers can lead to confusion as to which symbols represent given information and which represent unknown desired answers. You can resolve this difficulty by remembering--as recommended above--to "keep the goal in sight".)
Define symbols with mnemonic names. If a problem involves a helium atom colliding with a gold atom, then define mh as the mass of the helium atom and mg as the mass of the gold atom. If you instead pick the symbols m1 and m2, you stand a good chance of mixing up the symbols and their meanings as you solve the problem. And if you don't define the symbols at all, but just begin throwing around m's and M's, you'll confuse both yourself and whoever is grading your answer.
Keep packets of related variables together. In acceleration problems, the quantity (1/2)at2 comes up over and over again. This collection of variables has a simple physical interpretation, transparent dimensions, and a convenient memorable form. In short, it is easy to work with as a packet. Take advantage of this ease. Don't artificially divide this packet into pieces, or write it in an unfamiliar form like t2a/2. Packets like this come up in all aspects of physics--some are even given names (e.g. "the Bohr radius" in atomic physics). Look for these packets, think about what they are telling you, and respect their integrity.
Neatness and organization. I am not your mother, and I will not tell you how to organize either your dorm room or your problem solutions. But I can tell you that it is easier to work from neat, well-organized pages than from scribbles. I can also warn you about certain handwriting pitfalls: Distinguish carefully between t and +, between l and 1, and between Z and 2. (I write a t with a hook at the bottom, an l in script lettering, and a Z with a cross bar. You can form your own conventions.) These suggestions on neatness, organization, and handwriting do not arise from prudishness--they are practical suggestions that help avoid algebraic errors, and they are for your benefit, not mine. (On the other hand, it doesn't hurt to be neat and organized for the benefit of your grader. One course grader of mine pointed out: "If I can't read it, I can't give you credit.")
Avoid needless conversions. If the problem gives you one length in meters and another in inches, then it's probably best to convert all lengths to meters. But if all the lengths are in inches, then there's no need to convert everything to meters--your answer should be in inches. In fact, you might not actually need to convert. For example, perhaps two lengths are given in inches and the final answer turns out to depend only on the ratio of those two lengths. In that case, the ratio is the same whether the lengths going into the ratio are inches or meters. It's easy to make arithmetic errors while doing conversions. If you don't convert, then you don't make those errors!
Keep it simple. I will not assign baroque problems that require tortuous explanations and pages of algebra. If you find yourself working in such a way, then you're on the wrong path. The cure is to stop, go back to the beginning, and start over with a new strategy. (Generations of students have kept track of this rule by remembering to KISS: Keep It Simple and Straightforward.)
Answer Checking
Checking your answer does not mean comparing it to the answer in the back of the book. It means finding the characteristics of your answer and comparing them to the characteristics that you expect. Some of your problems--particularly the ones assigned early in the course--will actually lead you through the checking stage in order to familiarize you with the process. Other problems will leave it to you to perform this check. In either case, checking your answer is not just good problem solving practice that helps you gain points on problem assignments and on exams. The checking stage builds familiarity with the content of physics and the character of problem solutions, and hence develops your intuition to make solving other problems--and learning more physics--easier. (See Daniel F. Styer, "Guest comment: Getting there is half the fun", American Journal of Physics 64 (1998) 105-106.)
Dimensional analysis. Suppose you find a formula for distance (in, say, meters) in terms of some information about velocity (meters/second), acceleration (meters/second2), and time (seconds). If your formula is correct then all of the dimensions on the right hand side must cancel so as to end up with "meters".
Numerical reasonableness. If your problem asks you to find the mass of a squirrel, do you find a mass of 1,970 kilograms? Even worse, do you find a mass of -1,970 kilograms?
[Reasonable speeds. "My calculations give me a speed of 23 m/s. Is this reasonable?" It's hard for most people to get a feel for the reasonableness of speeds expressed in meters per second. Until this qualitative feel develops, Americans should check for reasonableness by converting speeds in meters per second to speeds in miles per hour: simply double the number (20 m/s is about 40 mi/hr). Non-Americans should convert to kilometers per hour: simply quadruple the number (20 m/s is about 80 km/hr).]
Algebraically possible. Would evaluating your formula ever lead you to divide by zero or take the square root of negative number?
Functionally reasonable. Does your answer depend on the given quantities in a reasonable way? For example, you might be asked how far a projectile travels after it is launched at a given speed with a given angle. Common sense says that if the initial speed is increased (keeping the angle constant) then the distance traveled will increase. Does your formula agree with common sense?
Limiting values and special cases. In the projectile travel distance problem mentioned above, the range is obviously zero for a vertical launch. Does your formula give this result? If you solve a problem regarding two objects, does it give the proper result when the two objects have equal masses? When one of them has zero mass (i.e. does not exist)?
Symmetry. Problems often have geometrical symmetry from which you can determine the direction of a vector but not its magnitude. More often they have a "permutation" symmetry: If your problem has two objects, you can call the cube "object number 1" and the sphere "object number 2" but your final answer had better not depend upon how you numbered your objects. (That is, it should give the same answer if every "1" is changed to a "2" and vice versa.)
Specify units. "The distance is 5.72" is not an answer. Is that 5.72 miles, 5.72 meters, or 5.72 inches? Similarly, if the answer is a vector, both magnitude and direction must be specified. (The direction may be drawn into a diagram rather than stated explicitly.)
Significant figures. Any number that comes from an experiment comes with some uncertainty. Most of the numbers in this course come with three significant figures. If a ball rolls 3.24 meters in 2.41 seconds, then report its speed as 1.34 m/s, not 1.34439834 m/s. Most introductory physics courses do not require a formal or technical error analysis, but you should avoid inaccurate statements like the second quotient above.
Large problems. If you break up your large problem into several subproblems, as recommended above, then check your results at the end of each subproblem. If your answer to the second subproblem passes its checks, but your answer to the third subproblem fails its checks, then your execution error almost certainly falls within the third subproblem. Knowing its general location, you can quickly go back and correct the error, so its effects will not propagate on to the remaining subproblems. This can be a real time-saver.
Summary
The problems in your physics course can be fun and exciting. Approach them in the spirit of exploration and they will not disappoint you!
Strategy design
Classify the problem by its method of solution.
Summarize the situation with a diagram.
Keep the goal in sight (perhaps by writing it down).
Execution tactics
Work with symbols.
Keep packets of related variables together.
Be neat and organized.
Keep it simple.
Answer checking
Dimensionally consistent?
Numerically reasonable (including sign)?
Algebraically possible? (Example: no imaginary or infinite answers.)
Functionally reasonable? (Example: greater range with greater initial speed.)
Check special cases and symmetry.
Report numbers with units specified and with reasonable significant figures
Sunday, March 23, 2008
'Weekends Are for Drinking' and Other Things Not to Say in Your Resume
You’ve thought long and hard about how to make your resume stand out, but perhaps you’ve gone too far. A recent survey conducted by Harris Interactive for CareerBuilder.com asked hiring managers to share the most unusual resumes they’ve seen. Explaining that you work well in the nude? What they found might surprise you.
CareerBuilder.com Survey: Top 12 Resume Disasters
1. Candidate mentioned in his resume that he spent summers on his family’s yacht in Grand Cayman.
2. Candidate attached a letter from his mother.
3. Candidate used pale blue paper with teddy bears around the border.
4. Candidate explained a gap in employment by saying he was getting over the death of his cat for three months.
5. Candidate specified that his availability was limited because Friday, Saturday and Sunday were “drinking time”.
6. Candidate included a picture of herself in a cheerleading uniform.
7. Candidate drew a picture of a car on the outside of the envelope and said it was the hiring manager's gift.
8. Candidate’s hobbies included sitting on a levee at night watching alligators.
9. Candidate included the fact that her sister once won a strawberry-eating contest.
10. Candidate explained that they worked well in the nude.
11. Candidate explained an arrest by stating, “We stole a pig, but it was a really small pig.”
12. Candidate included a family medical history.
“Anytime it’s beyond job function, it can be goofy”, said Jennifer Sullivan, senior career advisor at CareerBulider.com The survey also found people who listed bingo and smoking as their hobbies. “Others reference grade school classes and included old pictures,” Sullivan said.
CareerBuilder says to keep three key words in mind when writing your resume: simple, bold and professional. Instead of using stationery with bears, keep it clean and easy to read. The firm also recommends that you ask someone else to review it.
“You want to make yourself memorable for the right reasons,” said Sullivan. “Know your audience, and paint a picture of what you’ve done for other companies.”
Wednesday, March 19, 2008
The Milgram Experiment
A lesson in depravity, peer pressure, and the power of authority
The aftermath of the Holocaust and the events leading up to World War II, the world was stunned with the happenings in Nazi German and their acquired surrounding territories that came out during the Eichmann Trials. Eichmann, a high ranking official of the Nazi Party, was on trial for war crimes and crimes against humanity. The questions is, "Could it be that Eichmann, and his million accomplices in the Holocaust were just following orders? Could we call them all accomplices?"
Stanley Milgram answered the call to this problem by performing a series of studies on the Obedience to Authority. Milgram's work began at Harvard where he was working towards his Ph.D. The experiments on which his initial research was based were done at Yale from 1961-1962.
In response to a newspaper ad offering $4.50 for one hour's work, an individual turns up to take part in a Psychology experiment investigating memory and learning. He is introduced to a stern looking experimenter in a white coat and a rather pleasant and friendly co-subject. The experimenter explains that the experiment will look into the role of punishment in learning, and that one will be the "teacher" and one will be the "learner." Lots are drawn to determine roles, and it is decided that the individual who answered the ad will become the "teacher."
The generator has 30 switches in 15 volt increments, each is labeled with a voltage ranging from 15 up to 450 volts. Each switch also has a rating, ranging from "slight shock" to "danger: severe shock". The final two switches are labeled "XXX". The "teacher" automatically is supposed to increase the shock each time the "learner" misses a word in the list. Although the "teacher" thought that he/she was administering shocks to the "learner", the "learner" is actually a student or an actor who is never actually harmed. (The drawing of lots was rigged, so that the actor would always end up as the "learner.")
At times, the worried "teachers" questioned the experimenter, asking who was responsible for any harmful effects resulting from shocking the learner at such a high level. Upon receiving the answer that the experimenter assumed full responsibility, teachers seemed to accept the response and continue shocking, even though some were obviously extremely uncomfortable in doing so.
Today the field of psychology would deem this study highly unethical but, it revealed some extremely important findings. The theory that only the most severe monsters on the sadistic fringe of society would submit to such cruelty is disclaimed. Findings show that, "two-thirds of this studies participants fall into the category of ‘obedient' subjects, and that they represent ordinary people drawn from the working, managerial, and professional classes (Obedience to Authority)." Ultimately 65% of all of the "teachers" punished the "learners" to the maximum 450 volts. No subject stopped before reaching 300 volts!
Milgram also conducted several follow-up experiments to determine what might change the likelihood of maximum shock delivery. In one condition, the touch-proximity condition, the teacher was required to hold the hand of the learner on a "shock plate" in order to give him shocks above 150 volts.
The most amazing thing to note from this follow-up experiment is that 32% of the subjects in the proximity-touch condition held the hand of the learner on the shock plate while administering shocks in excess of 400 volts! Further experiments showed that teachers were less obedient when the experimenter communicated with them via the telephone versus in person, and males were just as likely to be obedient as females, although females tended to be more nervous.
Milgram's obedience experiment was replicated by other researchers. The experiments spanned a 25-year period from 1961 to 1985 and have been repeated in Australia, South Africa and in several European countries. In one study conducted in Germany, over 85% of the subjects administered a lethal electric shock to the learner.
Friday, March 14, 2008
Thursday, March 13, 2008
Wednesday, March 5, 2008
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