## Physical Science Tutorial

#### Intro

This document presents a step-by-step approach that you should find helpful in
solving problems typically assigned in engineering and science courses. The approach is illustrated with a few examples.

#### Sample Problem

1. Read the problem carefully. Understand what is given and what is asked for.
2. Make a sketch of the problem – this is very useful and should not be neglected!
3. Label the sketch with known information using a standard or meaningful symbols and their values; include units (eg, w = 50 lbs). Use subscriptions to differentiate multiple instances of the same quantity (eg, Ptop and Pbottom)
4. Label the sketch with asked-for information using a symbol and a “?”; include units
(e. g., F = ? N). Add any additional unknowns that you might think are relevant
5. Choose a unit system (either mks or fps) to work in based on the units that are given and/or required. It is wise to convert all given values into that unit system right at the beginning, so you won’t make the mistake of forgetting to do it later.
6. Think about and explicitly state any simplifying assumptions which can make the problem simpler. For example, are there aspects in the problem will have a negligible effect on the answer? Can these be ignored without affecting the answer significantly? Are there certain variables whose variation with some other variable that is so slight that it can be ignored? (ie, those variables can actually be assumed constant). For example, the density of water decreases with increasing temperature, but for most engineering applications, the variation can be ignored. Other common assumptions are steady state (unchanging with time) or an idealized shape (circle, cylinder, rectangle)
7. Now for the heart of the approach: finding a way to get from what you know to what you are asked to find. Think of and write down one or more equations that relate the asked-for information to at least some of the given information. Keep in mind that A) everything you need to know to solved for the asked-for variable may not be given (you may need to calculate a needed value as an intermediate result), and B) some of the given information may not be needed to solve the problem (i.e., there may be extra information given, so don’t worry too much if you are not using everything that is given).
It may help to write down in words your general approach to solving the problem. If you are not sure how to solve the entire problem, better to plan your approach than to jump into writing equations without knowing where you are headed. Writing out your approach will also help keep from getting confused about what to do next as you solve the problem.
8. If you know the values of all the variables in your equation except for the asked-for variable, you are in a good position. At this point, you can take one of two approaches:
a. solve the equation for the asked-for variable in symbolic terms, then substitute the values in for the symbols and calculate the answer, or
b. substitute values immediately on both sides of the equation and simplify, then solve of the asked for variable and simplify again.
The first method (keeping your equations in symbolic form until the very end) makes your solution is probably preferred by your professor: it is more understandable and is useful for application to future, similar problems. However, this method introduces many chances for error because you have to keep track of multiple terms, exponents and signs. The second method can make things simpler, reducing multiple terms to a single value.
Either way, sometimes, it may be impossible to explicitly solve the equation for the asked-for variable (ie, you cannot isolate the variable by itself on one side of an equation); in that case, you will have to resort to a trial-and-error solution.
8. If you do not know the values of all the variables in your equation except the desired unknown, then you must find a way to determine these values (or use a different equation). Think of more equations involving what you know and/or the unknown variables in the equations you have already written.
For example, say you are trying to find the volume of a cylinder with a diameter (D) of 10 cm and a height (H) of 25 cm. First, draw a sketch. You know the equation for volume (V) is V = H * A where h is height and A is the area of the end. But, you are given the diameter, not the area of the end. So, you need to come up with another equation, namely one that relates diameter to area – but this equation is not given. You know that the equation for the area of a circle is pi * radius2 and you know that diameter = 2 * radius. Combine these two equations you to get from diameter to area:

```	A = π r2
D = 2 r
solve for r: 		r = D/2
substitute into equation for A and simplify: 	 A = π (D/2)2  = πD2/4
substitute into equation for V and evaluate:
V = H*A = H*πD2/4 = 25 cm * π(10 cm)2/4 = 1962 cm3 ≅  2000 cm3
```

9. Include units on your intermediate and final answers – checking that units are correct will alert you if you made a mistake. Remember, to add or subtract terms, their units must be identical. If the units are different, you must have made a mistake.
11. Review your work. See if you dropped or transposed any variables, coefficients or exponents along the way. Also, it’s easy to get a + or – sign wrong, so check they are correct.
12. Check significant digits in the final answer by looking at the given values. Your final answer should be no more precise than the least precise given value. Any more than 4 digits would rarely be justified in most engineering problems. Three (or even two) is more commonly justified in engineering.
13. Check that your answer is correct by, if possible, inserting the answer into the original equation and verify that right-hand side equals the left-hand side. If they aren’t equal, you made a mistake somewhere. Try your check again first!

TIPS FOR PROBLEM SOLVING
1. Always draw a sketch! You must understand the problem to solve it and a sketch will help make sure you understand it.
2. Present your work neatly and organized. Lay out your work sequentially left to right, top to bottom on the page. If you are using two columns, draw a line to separate them. Don’t jump around on the page. If you run out of room and having to write something non-adjacent to where it appears next, draw an arrow to link them. Don’t draw more than a couple arrows like this, tho, or it will get very confusing. If your professor cannot understand what you are doing, you will get a lot of points off.
3. Include units when you substitute values; if the units do not work out correctly, that means you made an error. If you don’t include the units, you won’t be able to catch this error.
4. Annotate equations with a few words to explain what you are doing. It will help your professor understand what you are doing, which will help if you make a math error or your work is otherwise confusing. If your explanation is correct, you’ll get more partial credit if you make a math error because you professor will know that you knew what you were doing.
5. Don’t provide more than what is asked for. You won’t get any extra credit so it just gives more opportunities for a mistake that might cost you.

Example
You work for a company that makes rod-shaped products. You have been charged with streamlining the shipping process. You have found that the shipping folks waste a lot of time trying out what box to use to ship a rod of a given length. The boxes comes in a series of fixed sizes. You want to develop a table that relates the box dimensions to the longest rod that can fit in it so the shipping department can quickly figure out which box to use. Box A has dimensions of length = 50 cm, width = 20 cm and height = 20 cm; Box B has dimensions of length = 100 cm, width = 40 cm and height = 40 cm;
Find the longest rod that can be shipped in both Boxes.
(Adapted from Polya, 1945, and others.)

#### Solution

If you draw a sketch of a rectangularly-shaped box and think about you could position the longest rod in it, you can see that it should be placed diagonally both horizontally and vertically. When you start writing equations, you can see you will have to apply Pythagorean theorem twice — once to get the length of the diagonal in the horizontal (along the bottom of the box) and then again to find the length of the diagonal slanted from the bottom-left corner to the upper-right corner (or vice versa).

Let x, y and z represent the length, width and height of the box and r represent the length of the rod

r^2 = x^2 + y^2 + z^2

If you neglect to make the sketch, arriving at the solution could be much more difficult.

Once you have this equation, you can easily compute r for any combination of x, y and z.
You can make a little spreadsheet to do so:

```	Box A	Box B
x	50	100
y	20	40
z	20	40
r	57.4	114.9
```