A, R, D. When the decimal point moves Left, you Add; and when the decimal point moves Right, you Deduct.
In most instances, you will find it convenient to reduce the numbers used to numbers between 1 and 10 times 10 to the proper power. Unless otherwise specified, all answers to problems using powers of ten will conform to that requirement.
Powers of Ten Added and Subtracted Before using powers of ten in mathematical operations, it will be beneficial to review a few more principles governing exponents:
If two or more numbers are written with the same base, their product is equal to the base raised to a power equal to the algebraic sum of their exponents.
31 × 35×33 = 31+5+8 = 312 If two numbers are written with the same base, their quotient is equal to the base raised to a power equal to the algebraic difference of their exponents (numerator's exponent minus denominator's exponent).
= 45-3 = 42 A factor may be moved from numerator to denominator or from denominator to numerator by changing the sign of its exponent. Thus we have +==33X13=32 The bases must be the same before numbers can be multiplied or divided by the addition or subtraction of their exponents. Thus, d' X b' cannot be combined because the bases (a and b) are not the same.
Note particularly that the rules specify algebraic addition and algebraic subtraction of the powers. Here are some summarizing examples:
37 X 3-11 = 37+ (-11) = 37-11 = 3-4 = 15X15 = 8113 = 17 = 7 58 56 = 58(6) = 58+6 = 514.
612-6013 = 68=51 Multiplication and division employing powers of ten may be performed in three simple steps as follows:
(1) Reduce all numbers to values between 1 and 10 multiplied by 10 to the proper power.
(2) Perform the indicated operations.
(3) Change the result to a number between 1 to 10 multiplied by 10 multiplied by 10 to the proper power.
COMPUTING AREA Mensuration formulas deal with the dimensions, areas, and volumes of geometric figures. There are five geometric figures with which you should be familiar, and there is a separate formula for finding the area of each. The area of a plane figure is equal to the number of square units it contains. Areas are measured in different units as compared to measuring length. An area that is square and 1 inch on each side is called a square inch. All area units are square units, such as square inch, square foot, square yard, square rod, and square mile. Other area units are the square centimeter, the square meter, et cetera, found in the metric system of measurement.
TABLES OF AREAS 144 square inches (sq. in.) = 1 square foot (sq. ft.).
9 square feet = 1 square yard (sq. yd.).
30% square yards = 1 square rod (sq. rd.).
160 square rods = 1 acre (A).
640 acres = 1 square mile (sq. mile).
The technique for determining the area of any geometric shape is based upon the use of formulas.
To solve a problem by formula, it is necessary to-
(1) select the formula that covers the problem situation,
(2) insert the known values in the selected formula, and 15
AREA = LW S S AREA = S' S FIGURE 1-5. A rectangle.
(3) then make the necessary mathematical manipulations to find the unknown quantity.
The Rectangle A rectangle is a four-sided plane figure whose opposite sides are equal and all of whose angles are right angles (90°). The rectangle is a very familiar area in mechanics. It is the cross-sectional area oi many beams, rods, fittings, etc. (See figure 1-5.)
The area of a rectangle is the product of the measures of the length and width when they are expressed in the same units of linear measure.
The area may be expressed by the formula:
A = LW where: A = area.
L = length of rectangle.
W = width of rectangle.
EXAMPLE A certain aircraft panel is in the form of a rectangle having a length of 24 inches and a width of 12 inches. What is the area of the panel expressed in square inches?
FIRST: Determine the known values and substitute them in the formula.
A = LW A = 24 X 12 NEXT: Perform the indicated multiplication;
the answer will be the total area in square inches.
A = 24 X 12 = 288 sq. in.
FIGURE 1-6. A square.
The Square A square is a plane figure having four equal sides and four right angles (figure 1-6).
To determine the area of a square, find the product of the length of any two sides. Since a square is a figure whose sides are equal, the formula can be expressed as the square of the sides or:
A = S where A is the area and S is the length of a side.
• EXAMPLE What is the area of a square plate whose side measures 25 inches?
FIRST: Determine the known value and substitute it in the formula A = S A = 25% NEXT: Perform the indicated multiplication;
the answer will be the total area in square inches.
A = 25 X 25 = 625 sq. in.
Triangles A triangle is a three-sided polygon. There are three basic types of triangle: scalene, equilateral or equiangular, and isosceles. A scalene triangle is one in which all sides and angles are unequal, whereas the equilateral triangle, being just the opposite, has equal sides and equal angles. A triangle that has two equal sides and angles is known as an isosceles triangle.
Triangles may be further classified as to right, obtuse, or acute. These terms are descriptive of the included angles of the triangle. A right triangle is one that has one angle measuring 90°. In an obtuse triangle, one angle is greater than 90°, 16
SCALENE EQUILATERAL OR ISOSCELES Unequal Sides and EQUIANGULAR Angles Equal Sides and Angles Two Equal Sides and Angles
RIGHT RIGHT OBTUSE Right 4 - one L = 90° Obtuse A - one L > 90° Acute 4 - all 4 < 90°
ACUTE ACUTE FIGURE 1-7. Types of triangles.
while in an acute triangle all the angles are less than 90°. The various types of triangles are shown in figure 1-7.
The altitude of a triangle is the perpendicular line drawn from the vertex to the base. In some triangles, as in figure 1-8, it may be necessary to extend the base so that the altitude will meet it.
The base of a triangle is the side upon which 17
Circumference 2'6"
ALTITUDE Diameter d
- 3'2" BASE - FIGURE 1-8. Triangle.
the triangle is supposed to stand. Any side may be taken as the base.
The area of any triangle may be calculated by using the formula:
A = ab where A is equal to Area; § is a given constant;
a is the altitude of the triangle; and b is the base.
EXAMPLE Find the area of the triangle shown in figure 1-8.
FIRST: Substitute the known values in the area formula.
1 = jab = 1 = = × 26" × 32" NEXT: Solve the formula for the unknown value.
1140 1== × 30 × 38 = 1 = 570 sq. in..
Circumference and Area of a Circle To find the circumference (distance around) or the area of a circle it is necessary to use a number called pi (r). This number represents the ratio of the circumference to the diameter of any circle.
Pi cannot be found exactly because it is a neverending decimal, but expressed to four decimal places it is 3.1416, which is accurate enough for most computations. (See figure 1-9.)
Circumference The circumference of a circle may be found by using the formula:
FIGURE 1-9. A circle.
EXAMPLE The diameter of a certain piston is 5 inches.
What is the circumference of the piston?
FIRST: Substitute the known values in the formula, C = rd.
C = 3.1416 × 5 NEXT: Solve the formula for the unknown value.
C = 15.7080 inches.
Area The area of a circle, as in a rectangie or triangle, must be expressed in square units. The distance that is one-half the diameter of a circle is known as the radius. The area of any circle is found by squaring the radius and multiplying by ». The formula is expressed thus:
A = Tr?
where A is the area of a circle; 7 is the given constant; and r is the radius of the circle.
EXAMPLE The bore (inside diameter) of a certain aircraft engine cylinder is 5 inches. Find the cross sectional area of this bore.
FIRST: Substitute the known values in the formula, A = mr, 1 = 3.1416 X 2.52 NEXT: Solve the formula for the unknown value.
where C is the circumference; « is the given con- 4 = 3.1416 × 6.25 stant, 3.1416; and d is the diameter of the circle. 4 = 19.635 sq. in.
10' bz FIGURE 1-10. A trapezoid.
14" The Trapezoid FIGURE 1-11. Computing the area of a trapezoid.
A trapezoid (figure 1-10) is a quadrilateral having one pair of parallel sides. The area of a trapezoid is determined by using the formula:
1 = = (02 + 62) 12 where A is the area; ¿ is the given constant;
b and b are the lengths of the two parallel sides;
and h is the height.
EXAMPLE What is the area of a trapezoid whose bases are 14 inches and 10 inches, and whose altitude is 6 inches? (See fig. 1-11.)
FIRST: Substitute the known values in the formula.
#= (01+142) 2 1 = =(10 + 11)6.
NEXT: Solve the formula for the unknown value.
A = 2(21)% edge to trailing edge. If the wing is a tapered wing, the average width or chord, known as the mean chord, must be known in finding the area.
The formula for calculating wing area is:
A = SC where A is the area expressed in square feet, S is the wing span, and C is the average chord.
The process used in calculating wing area will depend upon the shape of the wing. In some instances it will be necessary to use the formula for finding wing area in conjunction with one of the formulas for the area of a quadrilateral or a
EXAMPLES
1. Find the area of the wing illustrated in To determine the area, it is necessary to decide what formula to use. It can be seen that the wing tips would form a 7-foot-diameter circle; the remainder of the wing planform is then in the shape of a rectangle. By combining the formulas for wing area and area of a circle, the area of a wing having circular tips can be calculated.
FIRST: Substitute the known value in the formula.
1 1 = 2X111 1 = 72 sq. in.
Wing Area To deseribe the planform of a wing (figure 1-12), several terms are required. Lo calculate wing area, it will be necessary to consider the meaning of the terms-span and chord. The wing span is the length of the wing from wing tip to wing tip.
The chord is the width of the wing from leading A = SC + TRE A = (25 - 7) (7) + (3.1416) (3.52).
The value for S is represented by the original wing span less the diameter of the circular tips.
NEXT: Solve the formula for the unknown value.
1 = (18 × 7) + (3.1416 × 12.25) 1 = 126 + 38.5 A = 161.5 sq. ft.
A = Wing Area, ft.* C = Average chord, ft.
S = Span, ft.
S - FIGURE 1-12. Wing planform.
2. Find the area of a tapered wing (figure 1-14) COMPUTING THE VOLUME OF SOLIDS whose structural span is 50 feet and whose mean Solids are objects with three dimensions-.
chord is 6'8". length, breadth, and thickness. They are of many FIRST: Substitute the known values in the shapes, the most common of which are prisms, formula. cylinders, pyramids, cones, and spheres. Occasionally, it is necessary to determine the volume A = SC of a rectangle, a cube, a cylinder, or a sphere.
A = 50' X 6'8". Since all volu les are not measured in the same NEXT: Solve the formula for the unknown value.
A = 50' × 6.67' A = 333.5 sq. ft.
3. Find the area of a trapezoidal wing (shown in figure 1-15) whose leading edge span measures 30 feet, whose trailing edge span measures 34 feet, and whose chord is 5 feet.
units, it is necessary to know all the common units of volume and how they are related to each other. For example, the mechanic may know the volume of a tank in cubic feet or cubic inches, but when the tank is full of gasoline, he will be interested in how many gallons it contains. The following table shows the relationship between some of the common units of volume.
UNITS OF SPACE MEASURE FIRST: Substitute the known values in the 1,728 cu. in. = 1 cu. ft.
formula. 27 cu. ft. = 1 cu. yd.
A = - (61 + b2) h 231 cu. in. = 1 gal.
7.5 gals. = 1 cu. ft.
1 = = (30 + 34)5. 2 pts. = 1 qt.
4 qts. = 1 gal.
NEXT: Solve the formula for the unknown value. Volume of a Rectangular Solid 1 A rectangular solid is a solid bounded by rec- 4 = 2(04)S tangles. In other words, it is a square-cornered 4 = 2(320) 4 = 160 sq. ft. 25' 6'8" 25' FIGURE 1-13. Wing with circular tips. FIGURE 1-14. Tapered wing with sweepback.
30 feet 5 feet 34 feet FIGURE 1-15. Trapezoid wing.
volume such as a box (figure 1-16). If the solid has equal dimensions, it is called a cube.
The formula for determining the volume of a rectangular solid may be expressed thus:
V = Iwh where: V = Volume.
1 = length.
w - width.
h = height.
EXAMPLE A rectangular-shaped baggage compartment measures 5 feet 6 inches in length, 3 feet 4 inches in width, and 2 feet 3 inches in height. How many cubic feet of baggage will it hold?
FIRST: Substitute the known values into the formula.
V = Loh V = 5'6" X 3'4" X 2'3".
NEXT: Solve the formula for the unknown value.
- S FIGURE 1-17. A cube.
If the rectangular solid is in the shape of a cube (figure 1-17), the formula can be expressed as the cube of the sides:
where Vis the volume and S is the side measurement of the cube.
Area and Volume of a Cylinder A solid having the shape of a can, length of pipe, or other such object is called a cylinder.
The ends of a cylinder are identical circles as shown in figure 1-18.
Surface Area The surface area of a cylinder is found by multiplying the circumference of the base by the altitude. The formula may be expressed as:
1 = Tah where A is the area, » is the given constant, d is the diameter, and h is the height of the cylinder.
V = A 105 = 41.25 cu. ft.
FIGURE 1-16. A rectangular solid. FIGURE 1-18. A cylinder.
Piston h = Stroke
- Cylinder
PISTON AT TOP CENTER PISTON AT BOTTOM CENTER FIGURE 1-19. Cylinder displacement.
EXAMPLE FIRST: Substitute the known values in the formula.
How many square feet of aluminum sheet would A - ndle be needed to fabricate a cylinder 12 feet long and 3 feet 6 inches in diameter? A = 3.1416 X 3'6" × 12'.
NEXT: Solve the formula for the unknown value.
A = 3.1416 X 3.5' X 12' A = 132.95, or 133 sq. ft.
Volume The volume of a cylinder may be found by multiplying the cross-sectional area by the height of the cylinder. The formula may be expressed as:
V = 7Th where V is the volume; » is the given constant;
r? is the square of the radius of the cylinder; and h is the height of the cylinder (figure 1-19).
EXAMPLE The cylinder of an aircraft engine has a bore (inside diameter) of 5.5 inches, and the engine has a stroke of 5.5 inches. What is the piston displacement of one cylinder? The stroke represents the height of the cylinder to be measured, cecause the volume displaced by the piston depends on the length of the stroke.
FIRST: Substitute the known values in the formula.
V = тґ°h V = (3.1416) (2.752) (5.5).
NEXT: Solve the formula for the unknown value.
V = 17.28 × 7.56 V = 130.64 cu. in.
GRAPHS AND CHARTS Graphs and charts are pictorial presentations of data, equations, and formulas. Through their use the relationship between two or more quantities may be more clearly understood. Also, a person can see certain conditions or relationships at a glance, while it would require considerable time to obtain the same information from a written description. Graphs may be used in a number of ways, such as representing a single equation or formula, or they may be used to solve two equations for a common value.
Graphs and charts take many forms. A few of the more common forms are called bar graphs, pictographs, broken-line curved-line graphs, and circle graphs. An example of each is shown in figure 1-20. The most useful of these graphs in technical work is the continuouscurved-line graph.
Interpreting or Reading Graphs and Charts It is more important, from the mechanic's viewpoint, to be able to read a graph properly than it is to draw one. The relationship between the horsepower of a certain engine at sea level and at any altitude up to 10,000 feet can be determined by use of the chart in figure 1-21. To use this type of chart, simply find the point on the horizontal axis that represents the desired altitude;
move upward along this line to the point where it intersects the curved line; then move to the left, reading the percent of sea level horsepower available on the vertical axis.
EXAMPLE What percent of the sea level horsepower is available at an altitude of 5,000 feet?
FIRST: Locate the point on the horizontal axis that represents 5,000 feet. Move upward to the point where the line intersects the curved line.
NEXT: Move to the left, reading the percent of sea level horsepower available at 5,000 feet. The available horsepower is 80%.
Nomograms It is often necessary to make calculations using the same formula, but using different sets of values for the variables. It is possible to obtain a solution by use of a slide rule or by preparing a table giving the solution of the formula resulting from successive changes of each variable. However, in the case of formulas involving several mathematical operations, the labor entailed would usually be very great.
It is possible to avoid all this labor by using a diagram representing the formula, in which each variable is represented by one or more graduated lines. From this diagram, the solution of the formula for any given variable may be read by means of an index line. A diagram of this type is known as a nomogram.
Much of the information needed to solve aeronautical problems will be presented in nomogram form. Instruction manuals for the various aircraft contain numerous nomograms, many of which 23
1955 ·→ 1960 +→→ 1965
(a) Bar graph (b) Pictograph
(c) Broken line graph (d) Continuous curved-line graph
(e) Circle graph FIGURE 1-20. Types of graphs.
SL 1 2 3 4 5 7 9 10 Altitude × 1000 ft:
FIGURE 1-21. Horsepower vs. altitude chart.
appear quite complex. Many of the presentations will possess several curves on the same coordinate axis, each curve drawn for different constants in the equation. In the latter case, it is essential to select the proper curve for the desired conditions.
Again, as with the simpler graphs, it is more important for the mechanic to be able to read nomograms than it is to draw them.
The following example is taken from the maintenance manual for the Allison 501-D13 turboprop engine. A nomogram (figure 1-22) is used to determine the power requirements when the engine is operating at minimum torque. The OAT (outside air temperature), station barometric pressure, and engine r.p.m. are three factors that must be known to use this particular nomogram.
EXAMPLE Determine the calculated horsepower of a certain engine, using the nomogram in figure 1-22.
Assume that the OAT is 10° C., the barometric pressure is 28.5 in. Hg, and the engine is operating at 10,000 r.p.m.
FIRST: Locate the reference points on the OAT scale and on the barometric pressure scale that correspond to the given temperature and pressure readings. These are identified as I and 2, respectively, on the chart. With the aid of a straightedge, connect these two points and establish point 3 on the pivot line.
NEXT: Locate the engine speed, identified as 1, on the engine speed r.p.m.
scale. Using a straightedge, connect points ® and Q and establish point $ on the calculated horsepower scale. The calculated horsepower is read at point 5. The calculated horsepower is 98%.
MODEL 501-D13 ENGINE AND MODEL 606 PROPELLER
CALCULATED TORQUEMETER
OUTSIDE AIR (READING) HORSEPOWER
•C TEMP •F PIVOT BAROMETRIC ENGINE SPEED RPM LINE PRESSURE - IN. HG
- 140 - 10500
32.0 -20 - 130 31.0 10400 NOTE: VALID FOR WINDS - 120 • 30.0 10 BELOW 11 M. P. H. 10300 Ễ110 29.0
- 10200 20 - 28.0 100 30 - 27.0 - 10100
• 26.0 · 10000 90 10 ~ 50 25.0 15-80 — 80 - 24.0 111 9800 20-70 23.0 25 - - 70
- 80
22.0 30 -
- 90 EXAMPLE 35 - - 21.0
- 100 - 80 40 -
- 110 - 55 - 20.0 FIGURE 1-22. Power requirements at minimum torque.