Engineering Statics Study Guide For Civil Engineering Technology Students
CHAPTER 1 FORCE VECTORS
1.1
Scalars and Vectors
Scalars and vectors can be looked at as “measurment” tools in mechanics. A scalar can be defined as positive or negative physical quantity possessing only magnitude (numerical value or size) like length, mass, temperature, and time. Where vectors can be defined as the quantities that require both magnitude and direction for a complete description. Examples of vectors we run into the subject of engineering statics are force, position, and moment. Here is a list of important information we need to establish regarding vectors. Referring to Figure (1.1): • A vector is shown graphically by an arrow • The length of the arrow represents the magnitude of the vector, and the angle (θ) between the vector and a fixed axis defines the direction of its line of action • The head or tip of the arrow indicates the sense of direction of the vector • Throughout this document, vectors will be represented by a letter with an arrow #» above it like ( A)
Figure 1.1: Vector components
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1.2
Vector Operations
Addition, subtraction, multiplication, and division can be applied to vectors. In the upcoming sections, we will look at the addition and subtraction of vectors. However, it is a good place to introduce the multiplication of a vector by a scalar first.
1.2.1
Vector Multiplication by a Scalar:
Referring to Figure(1.2) • Multiplying a vector by a positive scalar increases its magnitude that amount • Multiplying a vectore by a negative scalar changes the directional sense of the vector.
Figure 1.2: Scalar multiplication and division
1.2.2
Vector Addition:
If two “component” vectors are to be added to form a “resultant” vector, the addition can be carried out using the parallelogram law of addition or the triangle rule of addition. Parallelogram Law of Addition: Referring to Figure(1.3), we follow the following steps when adding two vectors shown in Figure(1.3a) using the parallelogram law of addition: • Join the tails of the component vectors as in Figure(1.3b) #» #» #» • Draw a line from the head of B parallel to A then draw a line from the head of A #» parallel to B • At the intersection, the point “P” completes and the drawn lines make the sides of a parallelogram. #» • R, which is the diagonal of the newly drawn parallelogram, represents the resultant #» #» #» vector R = A + B
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Figure 1.3: Parallelogram law of addition
Triangle Rule of Addition: Referring to Figure(1.4), we follow the following steps when adding two vectors shown in Figure(1.4a) using the triangle law of addition: #» #» • Join the head of vector A to the tail of vector B as in Figure(1.4b) #» #» • Draw a line from the tail of A to the tail of vector B to form the “resultant” vector #» R #» #» Note: Since addition is commutative, joining the head of vector A to the tail of vector B #» #» will yield the same results as joining the head of vector B to the tail of vector A because #» #» #» #» #» A+B =B+A = R
Figure 1.4: Triangle rule of addition
1.2.3
Vector Subtraction:
#» #» #» #» #» When subtracting vector B from vector A as R = A − B, the operation of subtraction #» #» #» can be represented as an addition as R = A + (−B). Graphically, that will reverse the #» direction of vector B and the Parallelogram law of addition as in Figure(1.5b) or the triangle rule of addition as in Figure(1.5c) can be applied here taking into consideration #» the new orientation of B.
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Figure 1.5: Vector subtraction
1.3
Forces as Vectors:
Forces can be fully expressed with magnitude and direction. Hence, they are vectors. As a result, the rules discussed in section (1.2.2) can be applied. Not only resultants can be found by adding forces, a resultant force can be resolved into components as well. basically, we work “backwards” from the Parallelogram law of addition or the triangle rule of addition to resolve the resultant force vector into components along an two axes to components as in Figure(1.6).
Figure 1.6: Force vector resolution
Note: Solving for resultant vector or resolving one will transfer the problem from vector addition (or subtraction) into “Trigonometry”. Therefore, a knowledge of such subject is a necessity in solving problems. Two important trigonometric laws should be presented here; the Law of Cosines and the Law of Sines. Referring to Figure(1.7): Law of Cosines: C=
p A2 + B 2 − 2AB cos (c)
Law of Sines: A B C = = sin (a) sin (b) sin (c)
Figure 1.7
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