In Chapter 7, I show you how to take multiple vectors and combine them into a single resultant behavior, which is a useful skill in helping simplify the number of actions on an object. That works fine if you're interested in examining the combined behaviors of an object, but what happens when you're interested in studying multiple behaviors but only have a single resultant to work with? For this situation, you need to understand how to create multiple behaviors of a resultant, or the components of a resultant vector.
The most useful feature of working with components is that these behaviors let you explore basic behaviors in more detail. For example, when an airplane is coming in for a landing, its approach is actually a vector with a given orientation at a specific speed. However, the pilot must maintain a certain horizontal behavior (which ensures that the plane actually reaches the runway and doesn't overshoot) to land the plane safely while guaranteeing that the vertical descent isn't so fast that it causes the plane to crash into a fiery heap when it hits the ground. The pilot needs to be aware of both the vertical and horizontal behaviors at the same time, for uniquely different reasons.
In this chapter, I show you how to break a single vector back into multiple behaviors that act entirely in Cartesian or non-Cartesian directions.
You may be asking yourself, "Why on earth would I care about breaking one combined action into several smaller actions?" Answer: In many statics and physics problems, you're often interested in examining these individual actions on a case-by-case basis. These individual actions of a single resultant vector are known as components. For example, consider the projectile motion example shown in Figure 8-1.
In this example, a player is shooting a basketball at a hoop in the distance. When she attempts the shot, the ball has a unique velocity (magnitude) and direction angle θ, which defines the initial path of the projectile (sense and line of action) at the exact instance she releases the ball. Together, these three pieces of information define the initial velocity vector of the basketball. (Flip to Chapter 4 for more on these three main vector properties.)
Because the player remains stationary as she attempts the shot (this isn't a slam-dunk attempt), the problem you want to examine is what's happening to the basketball itself — especially whether the path results in a scoring shot.
From experience, you should be able to recognize that as the player begins the shot, the ball moves upward with a certain vertical velocity while moving away from her with another horizontal velocity. These two velocities are the horizontal and vertical components of the combined (or resultant) velocity vector for the basketball.
In two dimensions, you need to represent two components for every resultant vector, and in three dimensions you actually have three components that you need to determine. However, for both two- and three-dimensional cases, these components together must always result in the same original combined behavior.
Note
A component of a resultant vector must also be the same type of vector as the resultant itself. If you're finding the components of a force vector, the components are also force vectors.
In the parallelogram method discussion in Chapter 7, I show you how to create a resultant vector of two vectors by constructing a basic parallelogram where the resultant is the diagonal across the parallelogram that shares the tail point of the original two vectors. The two sides of the parallelogram (the original vectors) that share the tail of the resultant are actually components of that resultant vector, as shown in Figure 8-2.
The process of creating a vector component is known as resolving a vector. In working with vectors in statics, you always resolve two-dimensional actions such as forces and displacements into exactly two pieces for two-dimensional problems and into exactly three pieces for three-dimensional problems. (You can also depict two-dimensional actions with three-dimensional components by making a third component that has zero magnitude.)
To accomplish this task, you need to decide which type of component is actually required. These types of components are often separated into two categories: Cartesian (or rectangular) components and non-Cartesian components.
Cartesian components: As the name Cartesian components implies, all of the resolved vector components are aligned with the Cartesian x-, y-, and (for three-dimensional problems) z-axes of your coordinate system. Sometimes you see them referred to as rectangular components. Rectangular components are probably the most common components you calculate and, fortunately, are usually the easiest to compute.
Non-Cartesian components: Non-Cartesian components of a vector aren't necessarily aligned with the Cartesian-axes. One or more components may be aligned with the Cartesian axes, but at least one is not. (If all were aligned with the axes, they'd be Cartesian components.)
In later chapters, I help you with the actual selection process for choosing the type of components to help you best solve a particular problem, and I give you some additional pointers on choosing the appropriate directions of your components. But for now, in the following sections I focus on explaining the specific calculation techniques you actually employ after you determine the type of component.
Note
After you compute the vector components, you can begin to compute their magnitudes, and represent them with vector equations.
To determine the components in two dimensions, you need to follow a few simple steps to create right triangles:
Form a right triangle from the vector by using a line parallel to the x-axis attached at the tail and a second line parallel to the y-axis and attached to the head of the vector.
Draw the x-component vector by drawing from the tail point of the original vector.
Locate the head of the x-component vector at the intersection of the horizontal and vertical lines.
Draw the y-component vector by drawing from the intersection of the horizontal and vertical lines (the tail point).
Place the head of the y-component vector at the head point of the original vector.
Figure 8-3 shows the finished diagram.
To determine the components in three dimensions, you use a similar head-to-tail construction technique.
Form a cube around the vector by placing one corner at the vector's tail and one at the opposite diagonal corner at the vector's head.
Establish reference lines by drawing a vertical line through the head point of the original vector; at the point where this vertical line crosses the horizontal plane, draw one horizontal line parallel to the x-axis and a second horizontal line parallel to the z-axis.
In the example in Figure 8-4, your first reference line is parallel to the y-axis and crosses the horizontal plane xz.
Determine the x-direction vector components by drawing the x- component vector from the tail point of the original vector.
In Figure 8-4, you place the point labeled Int. #1 at the head of the x- component, which represents the location where a horizontal line drawn parallel to the z-axis intersects with the x-axis of the reference coordinate system.
Establish the z-direction vector component by drawing the z-component vector from the head of the x-component to the point where the two horizontal lines intersect.
In this example, you draw from Int. #1 to Int. #2.
Determine the y-direction vector component by placing its tail at the horizontal intersection and its head at the head point of the original vector.
In Figure 8-4, you place the tail of the y-component vector at Int. #2.
For two-dimensional problems, the process of resolving a vector into components is as simple as drawing a right triangle. Consider the example in Figure 8-5a, which shows a force of 250 Newton acting at an angle of 130 degrees from the positive x-axis of the Cartesian coordinate system, which is the same as measuring 50 degrees from the negative x-axis.
To resolve this force vector into its rectangular components, you need to first locate a right triangle, indicated by the shaded region in Figure 8-5a. The original vector is located on the hypotenuse of this right triangle, and the component vectors are then drawn along the other two edges of the right triangle as shown in Figure 8-5b.
Using the principles of SOHCAHTOA (refer to Chapter 2), you can calculate the magnitude of the x-component of this from
where the angle 50 degrees is measured from the negative x-axis.
Similarly, you can compute the y-component,
Note
The values that you just computed are only the scalar magnitudes of the components and don't include the sense or direction of the component. For two-dimensional problems, you need to create component vectors or assign the directions by using simple logic and the Cartesian unit vectors i, j, and k, as I discuss in Chapter 5.
Using the bracket notation method of three-dimensional vector representation
Statics provides a few ways to represent vector information, including i, j, and k unit vector notations and scalar magnitudes and associated directions I discuss in Chapter 5.
But you'll see another bracket notation method pop up from time to time in calculus and physics textbooks, and it can be useful at times in statics. The major advantage of this notation is that the scalar components in each of the principle directions are readily visible and require no additional calculations to determine them.
For example, consider the acceleration vector a written in Cartesian notation:
The bracket representation for this vector is:
If you compare the Cartesian notation with the bracket notation, you see that each term in the bracket is actually one of the three scalar components of the Cartesian vector notation. In this acceleration example, the first term in the brackets, or the x-component, is +2.5 meters per second squared; the second term, or the y-component, is −3.1 meters per second squared; and the third term, or the z-component, is +6.2 meters per second squared.
In Figure 8-5, the x-component of the force has a magnitude of 160.69 Newton. However, looking at the vector representation of the x-component, you can clearly see that the component is acting to the left, or in the negative x-direction. Because you can denote the negative x-direction by using the –i unit vector, you can create the x-component force vector:
You calculate the y-component vector in a similar fashion. Recognizing that the y-component of this vector is acting in a positive y-direction, you can use a positive j unit vector to describe the direction.
Note
When you resolve a three-dimensional vector into its rectangular components, the components must be mutually perpendicular to each other, which usually means each component is parallel to one of the three Cartesian axes.
Determining the components of a three-dimensional vector is actually fairly simple as well. For example, if you wanted to find the rectangular or Cartesian components of the following velocity vector:
you simply need to strip off the numerical values that occur before the i, j, and k values in the vector representation.
Although Cartesian vectors always work with any statics problem you encounter, sometimes they aren't the most efficient tool in your proverbial toolbox.
The major advantage of using non-Cartesian components is that it allows you to use a more convenient coordinate system that may better match the symmetry of the object. For example, consider the force from a shockwave caused by an explosion. In this type of phenomenon, the shockwave moves in three dimensions, radially, away from the center of the explosion. If you use Cartesian representation to portray this force, you need three vector components for every small area on the surface of the shockwave, and each point on the surface has a different set of three components. However, if you chose to work in a different, non-Cartesian coordinate system such as spherical coordinates, you can transform this complex three-dimensional problem into a more simplified one-dimensional situation.
In this section, I show you how to calculate a non-Cartesian component and its corresponding Cartesian component. Figure 8-6 shows the same 250-Newton force from Figure 8-5 earlier in the chapter oriented at an angle of 130"degrees from the positive Cartesian x-axis (or shown as 50 degrees from the negative x-axis).
In this example, I've arbitrarily chosen to find one component in the direction of the Cartesian y-axis and the other, non-Cartesian component along the line Oa that is oriented 45 degrees below the negative x-axis. The following sections show you how you can use some of the resultant techniques from Chapter 7 to determine the magnitudes of these components.
In the parallelogram method, you basically construct a parallelogram with sides in the direction of the y-component and the component in the direction of Oa. Referring to Figure 8-6, you already know the resultant (250 Newton) and its direction (50 degrees from the horizontal). From geometry, you can conclude that the angle between Fy and FOa is 45 degrees as shown in Figure 8-7a. You can then pull the force triangle from the parallelogram and determine the angles geometrically as shown in Figure 8-7b.
With this information, you can find the components Fy and FOa from the law of sines.
From this relationship, you can then compute the scalar components:
An easier technique involves breaking each of these vectors into their respective x- and y- Cartesian components. First, you need to convert the resultant vector R into Cartesian form as shown in the following equation and Figure 8-8:
Even though you don't know the magnitudes of the components, you can still create a Cartesian vector form. Just keep the magnitudes in the equations as a variable:
Remember, you can find the resultant vector by performing vector addition (see Chapter 6):
You can then compare all the terms in front of the i unit vector, and all of the terms in front of the j unit vector to create a system of two equations that you can solve simultaneously.
and
Solving for the two unknown magnitudes gives you
Note
Components always come in pairs, and their magnitudes are directly related to each other. If you resolve a two-dimensional resultant into components, you must include both components in your calculations.
In this example, you calculated a Cartesian y-component and found it to be 352.21 Newton when paired with the non-Cartesian component along the line Oa. However, when you calculate the components for the same original force vector (in the earlier section "Using Cartesian concepts to calculate Cartesian components") the y-component only had a magnitude of 191.51 Newton for the same original force vector as this example. By comparison, these two y-components are significantly different. Depending on the direction of their counterpart components, two components acting in the same direction can vary quite dramatically. For this reason, you must always work with all of the components for a single force at the same time.