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From Wikipedia

How to Solve It

How to Solve It (1945) is a small volume by mathematician George Pólya describing methods of problem solving.

Four principles

How to Solve It suggests the following steps when solving a mathematical problem:

  1. First, you have to understand the problem.
  2. After understanding, then make a plan.
  3. Carry out the plan.
  4. Look backon your work.How could it be better?

If this technique fails, Pólya advises: "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"

First principle: Understand the problem

"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, Pólya taught teachers how to prompt each student with appropriate questions, depending on the situation, such as:

  • What are you asked to find or show?
  • Can you restate the problem in your own words?
  • Can you think of a picture or a diagram that might help you understand the problem?
  • Is there enough information to enable you to find a solution?
  • Do you understand all the words used in stating the problem?
  • Do you need to ask a question to get the answer?

The teacher is to select the question with the appropriate level of difficulty for each student to ascertain if each student understands at their own level, moving up or down the list to prompt each student, until each one can respond with something constructive.

Second principle: Devise a plan

Pólya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:

  • Guess and check
  • Make an orderly list
  • Eliminate possibilities
  • Use symmetry
  • Consider special cases
  • Use direct reasoning
  • Solve an equation

Also suggested:

  • Look for a pattern
  • Draw a picture
  • Solve a simpler problem
  • Use a model
  • Work backward
  • Use a formula
  • Be creative
  • Use your head/noggin

Third principle: Carry out the plan

This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals.

Fourth principle: Review/extend

Pólya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.

The book contains a dictionary-style set of heuristics, many of which have to do with generating a more accessible problem. For example:

The technique "have I used everything" is perhaps most applicable to formal educational examinations (e.g., n men digging m ditches) problems.

The book has achieved "classic" status because of its considerable influence (see the next section).

Other books on problem solving are often related to more creative and less concrete techniques. See lateral thinking, mind mapping, brainstorming, and creative problem solving.

Influence

  • It has been translated into several languages and has sold over a million copies, and has been continuously in print since its first publication.
  • Marvin Minsky said in his influential paper Steps Toward Artificial Intelligence that "everyone should know the work of George Pólya on how to solve problems."
  • Pólya's book has had a large influence on mathematics textbooks as evidenced by the bibliographies for mathematics education.
  • Russian physicistZhores I. Alfyorov, (Nobel laureate in 2000) praised it, saying he was very pleased with Pólya's famous book.
  • Russian inventor Genrich Altshuller developed an elaborate set of methods for problem solving known as TRIZ, which in many aspects reproduces or parallels Pólya's work.


From Yahoo Answers

Question:Two blocks of masses m1=1kg and m2=2kg are connected by a massless cord passing over a massless, frictionless pulley. The pulley is pulled up with a force of F=50 N. What are the accelerations of the two masses with respect to the ground?

Answers:Most important here is the direction of the acceleration. If the larger mass is on the right, then on that side acceleration is down. On the left, acceleration is up. Keep all forces that move in the direction of acceleration positive, opposite of it, negative. From Newton's 2nd law of motion, the net forces on the 1.0kg block (call it m )are: F = m a = T - m g + 50N-------->(1) The 2.0kg block has net forces of: F = m a = m g - T - 50N--------->(2) Adding (1) and (2) together and solving for acceleration: m a + m a = m g - m g a(m + m ) = g(m - m ) a = g(m - m ) / (m + m ) = 9.8m/s (2.0kg - 1.0kg) / (1.0kg + 2.0kg) = 3.3m/s (rounded) Hope this helps.

Question:http://tinypic.com/r/34dopwy/7 The mass show is hanging from light, smooth pulleys. It weighs 12.5 kg. How much force must be exerted in the direction shown to keep the mass stationary? I know the answer is 62.5, but I have no idea how to go about getting that answer.

Answers:if the rope and pulleys are massless, then the tension in the rope will be the same everywhere, so if you pull on the rope with a force T, all portions of the rope will exert a force T consider now the pulley supporting the weight; the rope on the left pulls up with a force T, and the rope on the right also pulls up with a force T the combination of forces must equal the weight of the mass, so we have 2T=W W=mg=12.5 kg x 9.8m/s/s = 122.5N T=61.25 N (if you are using g=10m/s/s, then you will get the answer that T=62.5N)

Question:Two blocks, m1 = 1 kg and m2 = 2 kg, are connected by a light string as shown in the figure***. If the radius of the pulley is 1 m and its moment of inertia is 5 kg m^2, What is the acceleration of the system? The figure is a pulley with m1 on the left side and m2 on the right side of the pulley. *** Figures cannot be pasted onto yahoo, so the link is http://www.scribd.com/doc/18103177/Serway-Physic-Chapter-11 It is problem number 12 (if you scroll down a bit), and the figure is on the right hand side. The answer is suppose to be (1/8)g. I would like to know how it was solved, I am using these problems as practice for the exam. Thank you in advance.

Answers:The net forces on m are: F = m a = T - m g------------------>(1) On m : F = m a = m g - T ------------------>(2) And on the pulley, the net torques are: = rF = I = r(T - T ) But angular acceleration is equal to a / r, so: Ia / r = r(T - T ) Ia / r = T - T ------------------------->(3) Adding (1), (2), and(3) together: m a + m a + Ia / r = m g - m g a(m + m + I / r ) = g(m - m ) a = g(m - m ) / (m + m + I / r ) = g(2kg - 1kg) / [(1kg + 2kg) + 5kg m / (1m) ] = g / 8 (one-eighth g) Hope this helps.

Question:I have troubles calculating the mechanical adavantage and need help answering some lab questions. 1) what are the units of % efficiency? (work divided by work=?) 2)how to calculate the mechanical advantage of a pulley system -3 fixed n 3 movable wheels? 3)what are the units of MA? 4)what is the relationship between mechanical advantage and the # of strands of string that are lifting a weight? 5) Since my pulley system has 3 fixed n 3 movable wheels, what should the input work be? if the work output is 0.1962 J?

Answers:1) Efficiency is a dimensionless quantity. It has no unit. 3) This mechanical advantage is also just a number. For the rest of the answers you should describe your pulley system better.

From Youtube

Physics - Acceleration on a Pulley :Solving for acceleration given two masses on a frictionless pulley.

Inclines Tension Pulleys :Watch the full video at: www.powerlearning21.com Introductory physics is full of problems involving two simple machines: inclines and pulleys. Inclines are often a student's first introduction to motion in two dimensions and serve as great examples of the utility of vectors. Pulleys are used to change the direction of a force, and require a new concept, tension. This lesson introduces tension and solves simple problems involving inclines and pulleys. An understanding of vectors and the laws of motion is required. Don't forget to subscribe, for new lessons uploaded on Power Learning 21- www.powerlearning21.com