# What is the answer? question 3

Today’s What is the answer blog has a dynamics question.  Here we go…

You are driving a car at 90 km/hr on a dry road and suddenly brake to stop.  If the coefficient of friction between the tires and the road is 0.63 what is the minimum stopping distance?

A. 34 m

B. 40 m

C. 51 m

D. 73 m

# Nomograms for the PE exam

## Nomograms: Antiquated technology or useful tool?

Many of today’s engineers would not consider using nomograms for the PE exam.  It would not surprise me if many of you reading this blog post have no idea what a nomogram (also called a nomograph) is.  This blog will quickly introduce them to you to help answer the question… Nomograms: Antiquated technology or useful tool?  Though they may be antiquated, you may find that they allow for very quick solutions to some types of problems.  Therefore, this blog is an introduction to the idea of using them for the PE exam to help you solve problems faster.

## What is a nomogram?

A nomogram is a graphical tool used to perform calculations, and they exist for numerous types of calculations.  They are essentially a diagram that gives a quick, approximate solution to a mathematical function.  A nomogram will have three or more lines (or curves), each for a different variable in the function.  The lines are placed on the nomogram to give the appropriate numerical relationships for the function.  If you know two or more of the variable values, you can use a straightedge to line up the values to graphically determine the unknown variable.  There will be some error, but it can be a very quick method for doing the calculations!

You can easily search the internet to get more detail on using (or making) nomograms.  There are also books teaching the lost art of nomography.  You can find books that discuss the history of nomography as well as numerous areas of application.  Omer Blodgett has a couple of great books, Design of Welded Structures and Design of Weldments, which have several great nomograms for topics in mechanics of materials.

## Here is a quick example

The figure below shows a nomogram used to solve the quadratic equation.  Though I did construct this particular nomogram I did not develop the idea (it has been around for a long time).  I am also not going to give details on how it was developed, but I will give it only as a representation of how they can be used.  The horizontal axis is the value of ‘a’ in the equation, the vertical axis is the value of ‘b’ in the equation, and the circle gives the two roots that solve the equation (values of ‘x’).  The dashed line shows an example with a = 1.5 and b = -2.  A line is drawn connecting those two points (the dashed line).  The roots are determined by the values on the circle where that line crosses.  The solutions for that particular example are x = 0.85 and x = -2.35, which are read from the values on the circle.

## These nomograms sound obsolete… are they really useful?

OK, I would agree that computers have made nomograms essentially obsolete.  However, you cannot use a computer during the PE exam.  I would also agree that the power of calculators make nomograms obsolete, but remember that you have a limited selection of acceptable calculators for the PE exam.  I am also not suggesting you use nomograms for everyday engineering design (though you could).  The real question is whether or not they can save valuable time during the PE exam.

You may not have a need for a nomogram to solve the quadratic equation, but they exist for far more complex problems.  Using them for the PE exam could allow for very rapid calculations for some common types of problems.  Machine Design Fundamentals by Hindhede, for example, provides a nomogram relating power, torsional stress, and shaft diameter that rapidly determines required shaft diameter.  Design of Welded Structures by Blodgett has rather complex nomograms that can be used to determine the required section modulus of beams (or required moment of inertia), which includes information about beam end conditions, beam length, beam loading, and allowable stress.  The same text also includes nomograms for deflection of curved beams, fatigue, torsional resistance, column effective lengths, and design aids for plate girders.  My textbook, Machine Analysis with Computer Applications (the same figure is in Mechanisms and Dynamics of Machinery by Mabie and Reinholtz), has a nomogram to determine maximum pressure angle in disk cams with roller followers.  I have also seen nomograms for fluid flow calculations to aid in pipe sizing.  All of these nomograms can save valuable time on the PE exam.  You are not limited by nomograms available in books… you can always try making some of your own for different calculations.

## Do you think nomograms are useful for the PE exam?

I hope you found this information useful.  The PE exam is all about working problems quickly and efficiently.  Nomograms can provide quick estimates of the answer (which is usually accurate enough for a multiple choice exam like the PE exam).  What are your thoughts?  Do you ever use nomograms for design?  Do you have recommendations of good nomograms to use for the PE exam (or good books that contain useful nomograms)?  Leave a comment letting everyone know and please share the information by tagging the share buttons.  Also, take a minute to go over to the Mechanical PE Academy Facebook page to get updates and exclusive content.

# What is the answer? question 2

Today’s What is the answer question comes from mechanics of materials.   Here we go…

A 1/2 inch diameter steel rod is stretched between two fixed points.  At a temperature of 72 degrees Fahrenheit the tensile load in the rod is 850 pounds.  At approximately what temperature (in degrees Fahrenheit) will the stress in the rod be zero?

A.  85

B.  95

C.  105

D.  115

# Four bar Mechanism Kinematics Shortcut

## Four bar Mechanism Kinematics Shortcut to save time on the PE Exam

For mechanical engineers the PE exam will definitely contain some dynamics problems.  If you choose to take the Mechanical Systems and Materials depth session (PM portion of the exam) you will also have problems related to machine analysis.  Both dynamics and machine analysis commonly contain problems related to the kinematics of four-bar mechanisms.  This blog will give some details on a cool four bar mechanism kinematics shortcut for calculating velocities for four bar mechanisms.  It is not a method that is frequently taught in dynamics or machine analysis courses, but it can definitely save some time on the PE exam.

## When can I use this shortcut?

A common type of problem has a four-bar mechanism with a given input angular velocity of the driving link.  The problem requires you to determine the angular velocity of the output link.  Even though this may seem fairly basic, the problem can be time consuming… and you don’t have a lot of time to waste during the PE exam.  This shortcut method is great for this type of problem!  This method also allows for a very quick estimation of the answer… which may be all you need on a multiple choice exam like the PE exam.

## What is the method?

OK… let’s look at this method.  The method is not new and has actually been around for a long time.  However, it is not commonly taught.  The method is based on the fact that the velocity vector of both coupler endpoints must have the same component along the long axis of the coupler.  Confusing?  Probably…  Let’s look at this in more detail with some illustrations.

## Terminology Review

Before I get into the method let me review some quick terminology for four-bar mechanisms.  The figure below shows a typical four-bar mechanism.  Link 1 is the ground link, which does not move.  We will call link 2 the driving link (or input link).  Link 3 is the coupler, and link 4 is the output link.

## Velocity Calculations

Now that we have reviewed the names of the links, let’s discuss how to calculate velocities using this method.  Say we know the input rotational velocity of the driving link (link 2).  The velocity of point A (part a of the figure below) can be calculated.  Link 2 is in pure rotation, so the velocity of point A is the product of the length of link 2 and the rotational velocity in radians per second.  Look now at part b of the figure.  Take the component of the velocity vector at A along the length of the coupler (parallel component).  That must also be the length of the component at B (equal parallel components).  The velocity at point B can easily be found as the vector perpendicular to link 4 with that parallel component.  The rotational velocity of the output link can be determined from that velocity if needed.

## Final Thoughts

The nice thing about this method for the PE exam is that it is a quick graphical method.  Drawing a simple sketch (roughly to scale) can allow you to estimate the velocity with minimal work.  Because the PE exam is a multiple choice exam, that quick estimate may be all that is required to get the answer.

I hope this quick introduction to this method helped you work mechanism problems more efficiently.  More details on this method, along with examples, are provided in my textbook Machine Analysis with Computer Applications (you can get it on Amazon here).  As a general note, the figures used in this post are from that textbook.

# What is the answer? question 1

What is the answer?  Thermodynamics question 1

Today’s question is a quick thermodynamics question.  Here we go…

A refrigerator has a coefficient of performance of 6.0 and the room temperature outside the refrigerator is 28 degrees Celsius.  The lowest temperature (in Celsius) that can be obtained inside the refrigerator is:

A.  5

B.  -10

C.  -15

D.  12

Work the problem and see if you can find the correct answer.  Give a comment below with the answer once you are done.  Also, share this with other so that more people get the practice.  Once you have done all of that… head on over to the Mechanical PE Academy Facebook page, where you will get updates and exclusive content.

# Introduction to the ‘What is the answer?’ blog series

For every pass I caught in a game, I caught a thousand in practice.”  ~ Don Hutson

Welcome to a new blog series… What is the answer?

Let me see if I can summarize the typical engineer wanting to take the PE exam.  You graduated from college and quickly started your first real engineering job.  Almost immediately you forget most of what you worked so hard to learn during college.  Your new job doesn’t require you to use most of your education, so you gradually forget even more.  You work for 4 years or so… and now you want to take the PE exam.  You grab a few books to start reviewing for the exam, and it is at that moment you realize that you don’t remember any of this!  Simple statics problems seem unbelievably difficult!  The only thing you remember about thermodynamics is that there is no such thing as cold… only absence of heat!  How are you going to relearn all of this material for the exam?

If that sounds anything like you… you are not alone!  Relearning all of that material must be a gradual process.  One must work a lot of problems to practice the material again.  How do you find the time to work a lot of problems?  That is where this blog series can help!

So what is this blog about?

The best way to relearn the material is to continually work a lot of problems!  Practice! How do you find the time to work a lot of problems?  That is where this blog will help you.  I will do the tedious process of coming up with questions.  All types of questions.  This is getting free practice exam questions!  They will cover all the different areas that are needed for the PE exam.  I will post a question along with 4 possible answers.  Your job… find the answer.  Post a comment with what you think the answer is.  This is a GREAT way to get practice throughout the week.  This is a GREAT way to learn from your peers working the same problem!  Give comments on how you solved the problem.  Start discussions about which answer is correct.  If you practice solving problems as part of your routine you will get faster at solving these problems.

What types of problems will be on the blog?

I will include all types of problems of different difficulty levels.  Some long problems and some short problems.  They will cover different topics in engineering and math.  You may think some questions are very easy… GREAT!  But remember… other people will find those questions difficult.  Everyone will have different areas of expertise and different starting points.  If you know how to work the problem help others figure out the solution.  Have fun with the process and participate in the extremely powerful peer learning!!

Let me know what you think!

I am excited about this series, but what are your thoughts?  What topics do you need the most help with?  As always, feel free to leave comments to help me develop this blog.  Also, take a second to share this blog with others so they can join in on the solutions.

# Integration by Parts – The Fast and Easy Way!

Work smarter, not harder

~Scrooge McDuck

# Integration by Parts

In this blog you will learn how to do integration by parts the fast and easy way!  First of all… the majority of problems on the PE exam will not require calculus, but it is possible.  Because it may be required… and because success on the PE exam is all about working problems quickly… I wanted to share this VERY SIMPLE and VERY FAST method for doing integration by parts!  In fact, it is so simple that you will wonder why this method is not commonly taught in calculus courses (I don’t know the reason either).  So even if you don’t need this method on the PE exam it is a great tool to have… and you can use it to impress your friends!

Why would I care… I don’t solve calculus problems by hand!

So you may be thinking… if any integration appeared on the PE exam I would look up the solution or use my calculator to solve the integral.  Though I do recommend you have a good math reference book for the exam (I highly recommend this one as a good overall math reference), I honestly believe that you could solve some basic integration by parts problems faster with the method I am about to show you because it takes a while to find the solution in the book.  As far as using the calculator, you must remember that the NCEES has a limited list of approved calculators.  So the calculator may not have all the functions for calculus.

What is integration by parts?

Before I get into the details of this VERY SIMPLE and VERY FAST method for doing integration by parts, I want to briefly remind you about the basic idea of integration by parts.  Integration by parts is based on the concept of the product rule of differentiation.  Integration by parts is a method commonly used when the function to be integrated is a product of an algebraic function and a trigonometric or exponential function.

OK… so let’s just get to the point!  What is this VERY SIMPLE and VERY FAST method?  The method presented here is one that I call the tabular method, because a simple table is developed to quickly develop the solution to the integral.  One nice advantage is that the solution is developed in one step regardless of the problem (no more repeated integration by parts in the same problem… which typically happens).

I am going to explain the process with an example.

I will illustrate the process by solving the integral shown.  The function is a product of an algebraic function and an exponential function, so integration by parts applies.

Step 1: Make the table

The first step is to construct a very simple table.  The first column is the term in the integral labeled u and the second column is the term labeled dv.  Differentiate the ‘u’ term in the first column and continue until the derivate becomes zero.  Integrate the ‘dv’ term and continue to the end of the first column.

Step 2: Cross-multiply and sum

The second step, which is the last step, is to cross-multiply and sum the results.  Before you do that you need to alternate positive and negative signs (+ and – as shown)… those will be used in the summation step.  Now simply multiply the terms as shown with the red arrows and complete the sum of all the terms (don’t forget the + and – signs).  Add the integration constant C and you are done!  Very fast, very simple, and all completed in one step!

Action Steps

What should you do now?  Go grab a calculus book and look for some example problems on integration by parts.  Solve those problems using this method to prove to yourself how much faster this process is compared to the standard method.

I hope this helped you improve your efficiency in integration!  I know it may be a little confusing from this blog post alone, so if you have questions or comments please let me know.  I can always add a couple more examples to help clarify if needed.  Also, please take a moment to share this post with others!  You can click any of the ‘share the knowledge’ tabs.  That helps me a lot!

# Mechanical PE Exam Topic Concepts Introduction

To acquire knowledge, one must study; but to acquire wisdom, one must observe.”

~ Marilyn vos Savant

Welcome to a new blog series on Mechanical PE Academy!  This blog is the overall introduction to a blog series to help you build your understanding of Mechanical PE Exam Topic Concepts… which is critical to passing the exam.

What is the overall intent of this blog series?

The main focus area for the Mechanical PE Exam Topic Concepts blog series is just as the name implies… it will focus on understanding the concepts of the different topics covered on the Mechanical PE exam.  Too many people preparing for the PE exam work problem after problem after problem and wonder why they do not pass the exam.  I believe the answer is generally that you focused on working problems and did not focus on understanding the concepts needed for solving the problems!  You cannot predict what problems you will need to solve on the PE exam.  Spending all of your study time on working problems only significantly helps you if the PE exam is filled with problems very similar (which will not happen).  You really need to understand the concepts!  You need to change your studying philosophy and ‘think outside of the box’ to focus more intently on understanding concepts.

So am I telling you to not work problems during your review?

No!  You need to work problems… lots of problems!  You should work all different types of problems.  What I am telling you is that you should not focus exclusively on working problems for the sake of working problems.  Don’t think that you have enough time during the PE exam to look through several books to find an example problem that will guide you through the solution for each problem.  Having a good grasp on the concepts will help you solve the problems quickly.

Here is a basic example…

Let’s say you are taking the PE exam and you need to solve a thermodynamics question asking about work done during an isobaric process.  If you are mostly unfamiliar with the concepts of thermodynamic processes you would need to grab your thermodynamics reference book and search for a similar style of problem to guide you through the process.  Understanding the concept, however, you would realize that an isobaric process is simply a process that has constant pressure.  Plotting that process on a p-v diagram and finding the area under the curve (which in this case is the VERY SIMPLE process of finding the area of a rectangle) will give you work done during the process.  Understanding the concepts can drastically expedite the solution process!!  You may think… that example is too easy… there would not be questions on the PE exam that only require me to solve for the area of a rectangle.  Honestly, you would be surprised how easy some questions can be if you have a good understanding of the concepts!  After all, the PE exam is testing you to see if you do understand the concepts!!

So here is the plan…

I plan to post blogs to this series frequently!  How frequently?  I am not sure yet… but frequently.  My goal is to help you understand the concepts so you can work problems faster!  I will try to give good actions for you to take after the blogs to help you reinforce your understanding of the concepts.  Check back often to get new topics.

What topics would you like to see discussed in this series?  Leave comments on ideas, and I will do my best to cover topics most needed.  Also, please share this post with others!  That helps me a lot.

# How many books should I take to the PE exam?

Never memorize something that you can look up.”

~ Albert Einstein

You probably don’t want to memorize all the information you learned during your studies to become a mechanical engineer.  Luckily you don’t need to for this exam, because the PE exam is an open-book exam.  Basically, you can take any books you want.  This blog is not intended to tell you which books to take to the PE exam… for my recommendations of reference books for the exam please go here.

So the question I get a lot… how many books should I take to the PE exam?  This is tough for me to answer, because it depends a lot on you… the test-taker.  I have seen people take the exam and only have a few reference books.  I have seen people with suitcase loads of books.  Though I don’t remember exactly, I think I used around 10 – 12 books and I had around 3 – 4 binders with miscellaneous information.  The biggest thing to remember here is that you don’t have a lot of time to search through several books to find an answer to any problem!!  Pick the minimal number of books that you feel covers the required material for the test.  Too many books will make it hard to find material.  If you have too few books you are risking not having the information you need.

A very useful tip is to pick the books you think you need when you start your studying, and use them as you prepare for the exam.  Store those books in a specific area so you know that is what you plan on using.  Over the time you prepare you will fine-tune the book selection.  You will determine that some books you thought you needed you actually never use while studying.  Modify your book selection as needed, and by the time you take the exam you will have optimized your book selection.

# What should I do the day before the exam?

~ William S. Burroughs

The day before the PE exam you will likely be in panic mode!  This is a huge test and there can be a lot of pressure to pass.  The big debate that is probably going on in your mind the day before the test… What should I do the day before the exam? … should I study or rest?  You can probably deduce my answer simply by observing the picture posted along with this blog.

I can remember, from my time as a college professor, walking into a classroom to give an exam.  I would look around the classroom and quickly observe the students.  Some were sitting calmly at their desk, with pencil and calculator ready, waiting on the test.  Others were franticly looking through the textbook and class notes to cram that last bit of information into their brain.  Based on that observation alone I know that different people answer the question of “What should I do the day before the test?” very differently.

My advice for the day before (maybe even longer) is to rest, though I know that may be easier said than done.  You have worked hard to get to this point.  If you planned correctly you had plenty of time to study.  Cramming information on the last day will not help, in my opinion.  I would recommend spending time relaxing with friends and family.  Go have a nice meal.  Clear your mind and relax.  You’re not going to suddenly forget all the information you spent months studying.  If you need to travel to take the exam, which is common, go the night before.  Don’t drive very far the day of!!  Spend the night relaxing in the hotel room.  Order room service and watch TV.  You deserve to be pampered a little!

If you absolutely must study the day before don’t overdo it!  I would suggest spending the time quickly reviewing and organizing rather than true studying.  If you start doing too much studying you will likely just get more stressed about the exam.  The exam is tough enough… don’t make it harder by having high stress levels.