Chapter 6 Length and Length-Related Parameters

Introduction In this chapter, we will investigate the role of length, area, and volume along with other length-related variables in engineering applications. Every physical object has a size. Some things are bigger than others. Some things are wider or taller than others.

Introduction You should develop an ability for guessing object size, because having a “feel” for dimensions will help you become a better engineer. If you decide to become a design engineer, you will find out that size and cost are important design parameters

There are different types of coordinate systems such as rectangular, cylindrical, spherical

Measurement of Length Early humans may have used finger length, arm span, step length, stick, rope and so on to measure the size or displacement of an object. Today, depending on how accurate the measurement needs to be and the size of the object being measured, we use other measuring devices, such as a ruler, a yardstick, and a steel tape.

Measurement of Length These devices are based on internationally defined and accepted units such as millimeters, centimeters, meters or inches, feet, or yards. For more accurate measurements of small objects, developed measurement tools such as the micrometer or the Vernier caliper, which allow us to measure dimensions within 1/1000 of an inch.

Measurement of Length In the last few decades, electronic distance measuring instruments (EDMI) have been developed that allow us to measure distances from a few feet to many miles with reasonable accuracy.

Measurement of Length The Global Positioning System (GPS) is another example of recent advances in locating objects on the surface of the earth with good accuracy.

Trigonometric Tools to analyze some size problems, let us review some of the basic relationships and definitions. For a right triangle, the Pythagorean relation may be expressed by: a2 + b2 = C2

Trigonometric Tools

Nominal Sizes versus Actual Sizes If you were to measure the dimensions of the cross section of a 2 x 4 lumber, you would find that the actual width is less than 2” inches (approximately 1.5 inches) and the height is less than 4” inches (approximately 3.5 inches). Manufacturers of engineering parts use round numbers so that it is easier for people to remember the size and thus more easily refer to a specific part. The 2 x 4 is called the nominal size . Other structural members are also given by nominal size which is different from the actual size such as: I-beams, pipes, tubes, screws, wires, and many other engineering parts. Examples of nominal sizes versus actual sizes are given in Table 7.2.

Nominal Sizes versus Actual Sizes

Radians as a Ratio of Two Lengths Consider the circular arc shown in Figure below. The relationship among the arc length, S, radius of the arc, R, and the angle in radians, θ, is given by

Strain as a Ratio of Two Lengths When a material (e.g. in the shape of a rectangular bar) is subjected to a tensile load (pulling load), the material will deform. The deformation,ΔL, divided by the original length, L, is called normal strain, as shown in Figure

Area= (Length x Width) [ unit: m2 ]

Let us now investigate the relationship between a given volume and exposed surface area. Consider a 1 m 1 m 1 m cube. What is the volume? 1 m3. What is the exposed surface area of this cube? 6 m2. If we divide each dimension of this cube by half, we get 8 smaller cubes with the dimensions of 0.5 m 0.5 m 0.5 m, as shown in Figure below. What is the total volume of the 8 smaller cubes? It is still 1 m3. What is the total exposed surface area of the cubes? Each cube has an exposed surface area of 1.5 m2, which amounts to a total exposed surface area of 12 m2.

Cross-sectional area Cross-sectional area also plays an important role in distributing a force over an area such as: foundations of buildings, hydraulic systems, and cutting tools. For example, why the edge of a sharp knife cuts well? What do we mean by a “sharp” knife? A good sharp knife is one that has a cross-sectional area as small as possible along its cutting edge. The pressure along the cutting edge of a knife is simply determined by

Area Calculations The areas of common shapes, such as a triangle, a circle, and a rectangle, can be obtained using the area formulas shown in the Table

Approximation of Planar Areas There are many practical engineering problems that require calculation of planar areas of irregular shapes. For these situations, you may approximate planar areas using any of the procedures such as 1- The Trapezoidal Rule The approximate the planar areas of an irregular shape with reasonably good accuracy using the trapezoidal rule. Consider the planar area shown in the Figure, by dividing the total area into small trapezoids of equal height h, as shown in the Figure. Then sum the areas of the trapezoids: A = A1 + A2 + A3 +…..+ An

3- Subtracting Unwanted Areas 2-Counting the Squares divide a given area into small squares of known size and then count the number of squares. This approach is shown in the Figure. Then need to add to the areas of the small squares the leftover areas, which may be approximated by the areas of small triangles. 3- Subtracting Unwanted Areas Sometimes, it may be advantageous to first fit large primitive areas around the unknown shape and then approximate and subtract the unwanted smaller areas. An example of such a situation is shown in Figure.

Example

Volume Volume is another important physical quantity, or physical variable. We live in a three-dimensional world, so it is only natural that volume would be an important player in how things are shaped or how things work. Volume= (Area x Height) [ unit: m3 ] For example, depending on the size of your car’s engine, its engine size is 3.8 liters, it is also safe to say that in order to fill the petrol tank you need to put in about 15 to 20 gallons (57 to 77 liters) of gasoline.

Volume Calculations The volume of simple shapes, such as a cylinder, a cone, or a sphere, may be obtained using volume formulas as shown in the Table.

Example for determination of volume

Example

Example

Example