Because you've chosen to study statics, you're probably already aware of the importance of mathematics in your studies. If not, I've got bad news: You just can't easily avoid numbers and calculations (particularly geometry and trigonometry) in your pursuit. Statics can provide you with solid physical principles for studying the world around you, but your skill with numbers and computations is what makes this information truly shine.
Think about the first rocket scientists that helped send astronauts to the moon with the Apollo missions in the 1960s and 1970s. Those scientists were among the smartest people on the planet at the time in physics theory, astronomy, dynamics, and statics (yes, even back then), among countless other areas of expertise. Imagine the success, or lack thereof, they would have experienced if they didn't have strong mathematical backgrounds.
This chapter reviews some of the mathematics skills required to efficiently solve statics problems. In this chapter I cover some basic nomenclature involving scientific notation, show algebra skills that prove useful, and review several geometric and trigonometric fundamentals. I conclude the chapter by showing how to use the power rule in calculus to integrate and differentiate.
In engineering, the accuracy of calculations can be the difference between a successful project and one that results in a pile of rubble on the ground. In all calculations, the numeric results have a certain number of digits that are meaningful, and some that serve no purpose other than to describe the magnitude of the number.
A significant digit is a nonzero value in any numeric quantity. Nonsignificant digits are those additional zeroes that help determine the magnitude of the number (such as the trailing zeroes on the number 2,500,000 or the leading zeroes on the decimal number 0.000156). The exception to this rule is the case of a zero digit that appears between two nonzero digits, such as in the number 106. In this case, the number 106 has three digits, all of which (including the zero) are considered significant.
Rounding is used to truncate irrational numbers. For example, the decimal form of π (pi) is 3.14159. . . . Because irrational numbers may have an infinite number of digits if you carry out the calculation far enough (which is the case of pi), you commonly round off the value to a specified number of places. For example, rounding 3.14159 . . . to two decimal places results in the value 3.14, which contains three significant digits. Similarly, rounding it to four decimal places gives the value 3.1416, or five significant digits.
Warning
Don't confuse significant digits with decimal places; the 3.14 estimation of pi contains three significant digits but only two decimal places.
One further complication in the accuracy of calculations involves figuring out how many significant digits you need in mathematical operations involving both extremely large numbers and very small numbers in the same calculation. Unfortunately, in many engineering calculations, you never know the number of significant digits you need to accurately represent a value until after you complete the calculation. However, if you keep a couple of basic rules of thumb in mind, you should be fine:
Multiplication and division: When multiplying two numbers that have a different number of significant digits, remember that the final result should have as many significant digits as the number with the smallest number of significant digits in the original calculation.
Addition and subtraction: When adding or subtracting two numbers that have different numbers of significant digits, remember that the final result should have as many significant digits as the number with the smallest number of significant digits in the original calculation.
Basically, no calculation result can have more significant digits than any of its original input values. For example, say you want to add these numbers:
123,456.789 + 0.000123456789 = 123,456.789123456789
The first value (123,456.789) contains 9 significant digits. The second value also contains 9 significant digits (remember, the preceding zeroes don't count). However, accurately reflecting the final sum of these two digits would require a staggering 18 significant digits to record precisely. But because 9 is the most significant digits in either term, that's the number you include in your answer.
Note
In this book, I try to carry at least three decimal places in all cases regardless of the number of significant digits involved in the calculation.
A popular system of reporting numerical quantities for engineers and scientists is scientific notation. This method of representing numbers is very useful for briefly stating numbers that are extremely large or small. Scientific notation uses the base power of ten to greatly shorten written numbers by using a combination of a numerical multiplier and a 10 raised to some exponential value.
By employing scientific notation, you eliminate a lot of unnecessary scribbling. For instance, suppose you want to measure the distance to Pluto (either in planet or non-planet form) from the Earth in miles. Now before I start to receive nasty e-mails from my astronomer friends, I concede that this distance is actually highly dependent on where both planets are located on their current orbital cycle (among other factors). But, for the sake of this argument, say that this distance is roughly 2.7 billion miles.
In regular notation, you can represent this number as 2,700,000,000 miles. Clearly, this large number of zeroes is unwieldy. Using scientific notation, you can simply report this number as 2.7 × 109 miles.
The first term is the numeric multiplier and contains all the nonzero terms of your number. You always place the decimal just to right of the first nonzero digit (meaning the multiplier never goes higher than the ones place). After the multiplication sign, the next term is always the 10's multiplier. To determine the exponential power (the little superscript number attached to the 10) on the multiplier, you need to count the number of decimal places required to move the decimal from its starting location in the original number to a position just to the right of the first nonzero numerical value. In this case, you move the decimal a total of nine spaces to the left (to land between the 2 and the 7), so the exponent is a 9.
Note
For much smaller numbers, you move the decimal to the right, which results in a negative exponent. For example, Planck's Constant (which is a value used to describe the size of quanta in quantum mechanics) is expressed as 6.62606 × 10−34 N · m · s. This number written in normal notation has 33 preceding decimal zeroes before the actual numerical values. Add the extra place you shift to move the decimal to the right of the first nonzero numerical value, and you have a total exponent of −34.
The other basic rule you need to keep in mind is that when multiplying two values expressed in scientific notation, you multiply the numerical multipliers, add the exponent portions, and then make any final adjustments to the decimal placement to insure that you have only one nonzero value to the left of the decimal place. When multiplying exponent values with the same base (the 10 in this example), you simply add the exponents.
Suppose you want to multiply the distance to Pluto by Planck's constant. (Honestly, I'm not sure when you'd need that calculation, but it's always good to be prepared!) First, you multiply the numerical multipliers and then compute the new tens exponents by adding them, as follows.
That comes out to 17.890362 × 10−25 miles · N · m · s. But you can have only one nonzero number to the left of the decimal point, so you have to adjust your answer to 1.7890362 × 10−24 miles · N · m · s. Of course, you should probably do some serious unit simplification on this answer — I cover that in Chapter 3.
In the world of statics, a few algebra skills can help you with some of the heavier lifting statics requires. In this section, I talk briefly about several common and useful algebra techniques you encounter in practice (and throughout this book). For more on these and other algebra topics, check out Mary Jane Sterling's Algebra I For Dummies and Algebra II For Dummies (Wiley).
One of the more convenient mathematical tricks you use in statics is determining the slope and equation of a line between two points. The slope of a line is the ratio of the change in elevation (or height) to the change in horizontal distance; you may know it more simply as "rise over run." The equation for slope (signified by m) is m=
For example, suppose that I rest a ladder on the ground at location Point 1 and on a ledge at location Point 2 (as shown in Figure 2-1) and want to define the properties of the line that connects these two points (the slope).
You may also remember that you can use slope to define the equation of the line passing through those two points, where b represents a constant for the y-axis intercept (or the point where the line crosses the y-axis at x = 0):
For the ladder example, you can solve for the numerical constant b by plugging in the values for either (x1, y1) or (x2, y2), both of which produce the same value for the constant. Note: arrow in the following equation just indicates that I'm skipping some basic math.
The constant b is dependent on where your measurement reference (or the origin — in this case, Point 1) is located. Thus, the equation of the line between Point 1 and Point 2 in Figure 2-1 is y = 0.5x + 0, simplified to y = 0.5x.
Calculating the slope proves to be a very handy trick when you deal with position vectors, which I cover in Chapter 5. After you have the equation of a line or function, you can make use of all sorts of cool mathematical tricks, which I point out in later chapters.
Sometimes the equation you have doesn't solve for the variable you need. In that case, you can use algebra to juggle the equation in a way that suits it to your needs.
Suppose, for example, that you're given the following equation for the moment of inertia (I) of a rectangle having width b and height h.
This relationship would be handy if you knew both b and h. However, suppose you know h and I and want to find b. Rearranging this equation produces a different equation that would be more helpful in this scenario:
Similarly, if you know b and I, you can rearrange the equation to solve for h:
In this book, I provide guidance on how to generically solve a statics problem to produce a final equation. After you have the general equation, you can always rearrange the terms (following proper mathematical protocol, of course), to solve for a specific variable, or to create completely new relationships.
Sigma notation (also sometimes called summation notation) is another popular form of shorthand notation; it utilizes the Greek symbol sigma (Σ), hence the name. Simply put, any time you see sigma notation, you know that you're about to do a whole lot of addition.
Suppose you have several variables (such as P1, P2, P3, and P4) that you want to add. You can simply write that equation as follows:
PTOTAL = P1 + P2 + P3 + P4
That isn't too bad an expression, so long as you don't want to add an extremely large number of terms. But what if you had a list of a thousand or a million terms? You'd need several sheets of paper and a whole lot of time to complete that problem! Imagine writing that expression as
PTOTAL = P1 + P2 + P3 + . . . + Pn – 1 + Pn
where n is equal to however many terms you have. Sigma notation allows you to conveniently express this type of equation in a single compact method:
This is sigma notation. The sigma indicates that this expression is an equation involving addition. The variable below the sigma represents a counter and is increased by one from the first term (in this example, 1) each time you add a term to the expression. The variable above the sigma indicates the value of the final term in the expression (or n in this example). If you wanted to add only the terms from P3 through Pn, you'd just change i = 1 to i = 3.
If you wanted to rewrite the first example of P1 through P4, you need only modify the variables above and below the sigma, again:
Pretty simple and compact, huh? In fact, in statics, the fundamental equations of equilibrium utilize sigma notation every time you write an equation. I explain more about equilibrium in Part V, but a word to the wise: Get familiar with sigma notation now, if you haven't already!
Algebra isn't the only basic math you encounter in statics (see the preceding section). You also use some basic geometry and trigonometry principles on a regular basis, so the following sections give you the lowdown on these concepts.
In this section, I introduce a couple of geometric relationships that show up frequently in statics. They show you how you can compute the total angles contained within a polygon as well as how to relate angles created by the crossing of multiple lines. Geometry For Dummies, 2nd Edition, by Mark Ryan (Wiley) gives you more detail on geometric concepts.
In any physical analysis problem, basic geometry often plays a very important role. Statics is no different — in many of the analysis problems you encounter, you need to make use of several basic geometric relationships.
The first relationship involves determining the total degrees in a polygon of a given number of sides. Triangles are very popular shapes within static analysis, as are quadrilaterals, parallelograms, and other higher-order shapes.
The sum of the interior angles for a polygon having n sides can be given by the expression
Total Degrees in Polygon = 180(n – 2)
You can easily confirm this formula by using your basic knowledge of triangles and quadrilaterals. A triangle has three sides (n = 3) and a total of 180 degrees: 180(3 – 2) = 180(1) = 180. Similarly, a quadrilateral has four sides (n = 4) and a total of 360 degrees. Quadrilaterals and triangles (such as those in Figure 2-2) account for the majority of the problems in this book, but on occasion you need to venture to more-complex shapes, and this rule proves handy for those cases.
Another relationship involves geometric constructions with two parallel lines, A and B, and a third line C that crosses them both (see Figure 2-3). In this figure, θ3 and θ5 are opposite interior angles (or angles that are across from each other whenever two lines cross), and θ4 and θ6 are also opposite interior angles.
In this construction, you can see that the angles θ1 and θ2 comprise the complete 180 degrees of line A. That is,
Similarly, along line B,
Along line C,
These three constructions thus imply that
One of the more common hang-ups I see when working with angular measurements is the basic confusion that exists between the units degrees and radians. It turns out that both of these base units are actually related to each other, as the following formula shows. Recall that a circular shape has 360 total internal degrees. In radians, this same internal angle is represented by 2π radians. (Remember that π = 3.14159. . . .)
Note
You definitely want to pay special attention to which unit setting your calculator is currently working with — most calculators are capable of dealing with both degrees and radians, and many calculators can be easily (and accidentally) switched between these two modes. In fact, I've experienced many a calculation going awry because I failed to switch modes. But if you're careful, this mix-up won't be a major issue. Be sure to consult with your calculator's instruction manual if you're having issues with switching between units.
The Pythagorean theorem is another useful geometric relationship that allows you to relate the sides of a right triangle (a triangle with one angle of exactly 90 degrees). Consider the right triangle shown in Figure 2-4.
You may have seen the formula written as C2 = A2 + B2 or H2 = A2 + O2. Regardless of which letters you use, this formula relates the hypotenuse (the side opposite of the 90 degree angle) to the two other sides. This formula is very useful when you when you start working with vector resultants (see Chapter 7).
Trigonometry is the branch of mathematics that deals with triangles. The cornerstones of trigonometry are the sine, cosine, and tangent functions that define the relationships among the sides of a right triangle. Referring to Figure 2-4, you can see that Side A is the side adjacent to the reference angle θ, and Side O is the side opposite to the reference angle. Finally, Side H represents the hypotenuse of the right triangle and is found directly across from the right angle.
Tip
To help you remember these relationships, try using the anagram SOHCAHTOA. No, Sohcahtoa wasn't the guide who helped Lewis and Clark explore the western frontier and ultimately discover the Pacific Northwest. SOHCAHTOA can, however, be a tremendous guide for remembering the three basic identities of trigonometry:
The hardest part is just remembering how to spell it! S-O-H-C-A-H-T-O-A!
Note
You want to be sure to carefully denote which angle of the right triangle is your reference angle, because its location can affect your assignment of O and A in those expressions.
A few of the basic calculus skills that may come in handy in your statics work include the differentiation and integration of polynomials and the locations and value of maximum and minimum values of polynomial functions. Luckily for you, I discuss both in the following sections. Check out Mark Ryan's Calculus For Dummies and Mark Zegarelli's Calculus II For Dummies (Wiley) for a complete calculus review.
Before I illustrate a few of the simpler basics of calculus, keep in mind that there is significantly more to differentiation in calculus than just the power rule. After all, most engineers and scientists are required to take multiple semesters (sometimes three or four) of various levels of calculus to complete their degrees.
That being said, a large portion of the content covered in a basic statics course can be encompassed with the power rule, so that's where I start.
The derivative of a function represents the slope of the tangent line to the function at a particular location. The derivative of a constant is always zero. For a simple function f(x), I define the derivative as f'(x), which is equivalent to df(x)/dx. (In this case, the derivative represents the slope of the tangent line to the function at x.) The power rule states that for a smooth and continuous polynomial of order n, the derivative of a function f(x) can be expressed as
Note
The order of a polynomial determines the shape of the curve. A zero order polynomial is constant, a first order polynomial is linear, and a second order polynomial is curved (or more specifically quadratic).
For example, for the function
f(x) = 4x3 + 5x2 + 24
you can compute the derivative of as
f'(x) = 4 · (3) · x(3 – 1) + 5 · (2) · x(2 – 1) + (0) · 24 = 12x2 + 10x
The terms inside the parentheses indicate the powers of the original term being differentiated. Because the derivative of a constant is always zero, the 24 in this equation disappeared.
Note
The examples I show here are for first derivatives, but you can also have higher-order derivatives, such as second, third, or even hundredth derivatives, in calculus. To compute the second derivative of a function, you compute the first derivative as I explain here and then compute the derivative of that derivative. The higher the order derivative that you want to compute, the more derivatives you have to take. Fortunately, in statics, usually a second or third order derivative is sufficient.
For basic integration, a definite integral for a simple polynomial can employ a reverse process to the differentiation technique for the power rule. The following equation assumes that the polynomial function f(x) is smooth and continuous and evaluated between an upper limit b and a lower limit a.
When you perform this calculation, you're actually finding the area under the function between the limits of a and b. This value can come in really handy when you start calculating centroids (see Chapter 11). To integrate a smooth and continuous polynomial of order n such that f'(x) = xn, the integral becomes
On many occasions, the statics equations you write contain variables that are frequently in the form of smooth and continuous polynomials (meaning that the graph of the function doesn't contain any jumps or sharp changes) of some order n.
This setup is pretty convenient because the power rule I discuss in the previous sections works effectively on polynomials. The ability to be able to determine the locations of maximum and minimum values of a polynomial function is even handier. If you recall that the slope of a line tangent to a maximum or minimum value is always horizontal (or equal to zero), you shouldn't be surprised that
In order to find the location of a local maximum or minimum value, all you need is the first derivative of the original function, the ability to set that first derivative equal to zero, and the ability to find the value(s) of the independent variable x that satisfy that equation. After you determine the locations, simply plug those x values back into the original function f(x) and compute the value of that function. For example, consider a third-order function:
f(x) = x3 + 5x2 – 8x – 12
Setting the first derivative equal to zero allows you to find the locations x of the local maximum and minimum values.
f'(x) = 3x2 + 10x – 8 = 0
For this equation, you can find that x1 = 0.667 and x2 = −4.000. Substituting these locations into the original function, you can determine which is the location of the local maximum value.
f(0.667) = −14.815 and f(–4.000) = 36.000
From this result, you can conclude that the local maximum value of the polynomial f(x) is +36.000 and occurs at a location of x = −4.000.