Force Vectors
ES 1022y Engineering Statics
Force Vectors
Force Vectors In this course forces (acting along the vector axis), moments (rotation about the vector axis), and position vectors (moving along the vector axis) are all vector quantities. A force, moment or position vector has:
A magnitude, a direction, a sense, a point of application A vector quantity can be represented graphically by an arrow that shows its magnitude, direction and sense. Magnitude: characterized by size in some units, e.g. 34 N; represented by length of the arrow according to some scale, say, 1 cm = 10 N → 3.4 cm = 34 N. Direction: the angle between a reference axis and the arrow's line of action. Sense: indicated by the arrowhead (one of two possible directions)
A Word on Vector Notation In the lecture notes and text book a vector quantity is indicated by a letter in boldface type (F), while the magnitude of a vector is denoted by an italicized letter (F). For handwritten work a vector is usually indicated by drawing an arrow above the letter representing the vector, thus
Similarly, unit vectors can be denoted in handwritten work by drawing a hat symbol above the letter to give
6
ES 1022y Engineering Statics
Force Vectors
Vector Addition Consider two force vectors A and B. We want to add them together to find the vector sum, or resultant force vector, R such that
R=A+B=B+ A
We can do this using one of two methods. Parallelogram law:
If the two forces A and B are represented by the adjacent sides of a parallelogram, then the diagonal of the parallelogram is equal to the vector sum of the two forces.
Triangle of forces: Special case of the parallelogram law.
7
ES 1022y Engineering Statics
Force Vectors
As a special case, if the two vectors A and B are collinear, that is
Both vectors have the same line of action the parallelogram law reduces to an algebraic or scalar addition
Where we have three or more forces we can either use repeated applications of the parallelogram law or a force polygon to find the resultant force.
We can also use trigonometry to add two force vectors together using the sine and cosine laws. Consider a triangle with sides of length A, B, and C, and corresponding interior angles a, b, and c.
Sine law:
Cosine law:
8
ES 1022y Engineering Statics
Force Vectors
Resolution of a Force Vector A force vector can be resolved into two components with known lines of action using the parallelogram law.
In cases where we need to determine the resultant of more than two forces it is often easier to resolve each force into its components along specified axes, before adding these components algebraically to find the resultant. In this case we usually resolve each force into components using a Cartesian coordinate system.
9
ES 1022y Engineering Statics
Force Vectors
Example Problem
The force F = 450 lb acts on the frame. Resolve this force into components acting along members AB and AC, and determine the magnitude of each component.
10
ES 1022y Engineering Statics
Force Vectors
Unit Vectors For a vector A with a magnitude of A the unit vector is defined as
The unit vector has:
The vector A can then be represented as
Cartesian Unit Vectors In a three-dimensional, rectangular Cartesian axis system the Cartesian unit vectors i, j, and k are used to designate the directions of the x, y, and z axes respectively.
The Cartesian unit vectors have a dimensionless magnitude of 1 and a sense that is given by either a plus or minus sign to show whether they are pointing along the positive or negative x, y or z axes.
11
ES 1022y Engineering Statics
Force Vectors
Two-Dimensional or Coplanar Forces Now consider a two-dimensional, or co-planar, force vector F, as shown.
Resolving the force into components acting along the x and y axes allows us to write the force F as
The magnitudes of each component of F are represented by the positive scalars Fx and Fy. These are often referred to as the rectangular components of F. If we define θ as the angle between the line of action of the force F and the positive x axis, then we can write
Force Addition Using Components Consider three coplanar forces (forces lying in the same plane) F1, F2, and F3.
12
ES 1022y Engineering Statics
Force Vectors
These can be represented using Cartesian vector notation as
The resultant force vector is then given by
In the general case we can write
It is important to remember to take sign conventions into account. Components along positive coordinate axes have positive values and vice versa.
13
ES 1022y Engineering Statics
Force Vectors
The magnitude of the resultant force vector is given by
while the angle is given by
Three-Dimensional Force Vectors In many situations we need to solve problems in three dimensions, rather than the two that we have considered so far. To do this we use a three-dimensional, rectangular Cartesian coordinate system that is said to be right handed. Consider a vector A in 3-D space.
.
The vector can be represented as
with a magnitude given by
14
ES 1022y Engineering Statics
Force Vectors
The orientation of A is defined by the coordinate direction angles α, β, and γ. These are measured between the tail of A and the positive x, y, and z axes respectively, and will always be between 0o and 180o.
The angles are defined by the direction cosines
Recall that in Cartesian vector form A can be written as
The unit vector in the direction of A is then
15
ES 1022y Engineering Statics
Force Vectors
This implies that the unit vector can be rewritten in terms of the direction cosines
Recall that the magnitude of a vector is obtained as
Therefore the magnitude of the unit vector is given by
Since
we can also write A in terms of the coordinate direction angles
If one of the coordinate direction angles is missing, we can always work out what it is by using the equation
and rearranging it to find the cosine squared of the missing angle. The only catch is that the cosine itself can either be positive (angle is less than 90o), or negative (angle is greater than 90o). To determine which is the correct choice we need to determine whether the component of the force acting along the axis associated with the missing angle is acting in the positive direction (cosine is positive and the angle is less than 90o), or the negative direction (cosine is negative and the angle is greater than 90o),
16
ES 1022y Engineering Statics
Force Vectors
As an alternative to using coordinate direction angles, the direction of a force A can also be described using two angles θ and , such as those shown above. In this example the components of A can be found by first resolving the force into vertical and horizontal components using the angle in the vertical plane, which yields
The angle θ lying in the horizontal plane can then be used to resolve the horizontal component of the resolved force into components acting along the x and y axes giving
Finally, force of A can be written in component form as
where the angles θ and are related to the coordinate direction angles α, β, and γ by the following expressions
The above equations should not be memorized; instead it is important to understand how trigonometry was used to determine the components.
17
ES 1022y Engineering Statics
Force Vectors
Example Problem
Determine the magnitude and coordinate direction angles of the resultant force acting on the bracket.
18
ES 1022y Engineering Statics
Force Vectors
19
ES 1022y Engineering Statics
Force Vectors
Example Problem
If the resultant force acting on the bracket is directed along the positive y axis, determine the magnitude of the resultant force and the coordinate direction angles of F so that β < 90o.
20
ES 1022y Engineering Statics
Force Vectors
21
ES 1022y Engineering Statics
Force Vectors
Position Vectors A position vector is a vector that locates a point in space relative to another point.
Consider a position vector extending from the origin O to a point P (x, y, z) in space. The position vector r of the point P relative to the origin O is then given by
In the more general case given a point A (xA, yA, zA) representing the tail of a vector and another point B (xB, yB, zB) representing the head of the vector, the position vector rAB is given by
22
ES 1022y Engineering Statics
Force Vectors
Force Vector Directed Along a Line If a force is directed along a line and the position of two points along the line is known the force vector can be represented in Cartesian coordinates as follows:
This is very useful if we are trying to represent a force vector acting along a cable with known starting and ending coordinates, remembering that the force in the cable will always be acting in tension (away from the point that the force is acting at).
23
ES 1022y Engineering Statics
Force Vectors
Example Problem
The antenna tower is supported by three cables. If the forces of these cables acting on the antenna are FB = 520 N, FC = 680 N, and FD = 560 N, determine the magnitude and coordinate direction angles of the resultant force acting at A.
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ES 1022y Engineering Statics
Force Vectors
25
ES 1022y Engineering Statics
Force Vectors
Dot Product
The dot product of two vectors A and B, written as A·B, is defined as the product of the magnitudes of A and B and the cosine of the angle θ between their tails. Mathematically it can be written as
Note that the result of the dot product is a scalar and not a vector, with units that are the product of the units associated with vectors A and B. Now consider the dot product of the Cartesian unit vectors i, j, and k.
If we now express vectors A and B in Cartesian form we can write
The dot product has two important applications in statics. Finding the angle between two vectors Given two vectors A and B, the angle θ between the two vectors can be found by
26
ES 1022y Engineering Statics
Force Vectors
Component of a force vector parallel to a particular direction or line
If the direction of the line is specified by the unit vector u, the projection of vector A onto the line is given by
Since the result is a scalar, the vector representation of A|| is given by
The magnitude of the component of vector A perpendicular to the direction or line can be found by using either or the theorem of Pythagoras while the vector form can be found from
27
ES 1022y Engineering Statics
Force Vectors
Example Problem
Determine the projected component of the force FAB = 560 N acting along the cable AC. Express the resultant as a Cartesian vector.
28
ES 1022y Engineering Statics
Force Vectors
29
ES 1022y Engineering Statics
Force Vectors
Example Problem
Determine the angle θ between the two cables attached to the post.
30
ES 1022y Engineering Statics
Force Vectors
31
Force Vectors
Force Vectors In this course forces (acting along the vector axis), moments (rotation about the vector axis), and position vectors (moving along the vector axis) are all vector quantities. A force, moment or position vector has:
A magnitude, a direction, a sense, a point of application A vector quantity can be represented graphically by an arrow that shows its magnitude, direction and sense. Magnitude: characterized by size in some units, e.g. 34 N; represented by length of the arrow according to some scale, say, 1 cm = 10 N → 3.4 cm = 34 N. Direction: the angle between a reference axis and the arrow's line of action. Sense: indicated by the arrowhead (one of two possible directions)
A Word on Vector Notation In the lecture notes and text book a vector quantity is indicated by a letter in boldface type (F), while the magnitude of a vector is denoted by an italicized letter (F). For handwritten work a vector is usually indicated by drawing an arrow above the letter representing the vector, thus
Similarly, unit vectors can be denoted in handwritten work by drawing a hat symbol above the letter to give
6
ES 1022y Engineering Statics
Force Vectors
Vector Addition Consider two force vectors A and B. We want to add them together to find the vector sum, or resultant force vector, R such that
R=A+B=B+ A
We can do this using one of two methods. Parallelogram law:
If the two forces A and B are represented by the adjacent sides of a parallelogram, then the diagonal of the parallelogram is equal to the vector sum of the two forces.
Triangle of forces: Special case of the parallelogram law.
7
ES 1022y Engineering Statics
Force Vectors
As a special case, if the two vectors A and B are collinear, that is
Both vectors have the same line of action the parallelogram law reduces to an algebraic or scalar addition
Where we have three or more forces we can either use repeated applications of the parallelogram law or a force polygon to find the resultant force.
We can also use trigonometry to add two force vectors together using the sine and cosine laws. Consider a triangle with sides of length A, B, and C, and corresponding interior angles a, b, and c.
Sine law:
Cosine law:
8
ES 1022y Engineering Statics
Force Vectors
Resolution of a Force Vector A force vector can be resolved into two components with known lines of action using the parallelogram law.
In cases where we need to determine the resultant of more than two forces it is often easier to resolve each force into its components along specified axes, before adding these components algebraically to find the resultant. In this case we usually resolve each force into components using a Cartesian coordinate system.
9
ES 1022y Engineering Statics
Force Vectors
Example Problem
The force F = 450 lb acts on the frame. Resolve this force into components acting along members AB and AC, and determine the magnitude of each component.
10
ES 1022y Engineering Statics
Force Vectors
Unit Vectors For a vector A with a magnitude of A the unit vector is defined as
The unit vector has:
The vector A can then be represented as
Cartesian Unit Vectors In a three-dimensional, rectangular Cartesian axis system the Cartesian unit vectors i, j, and k are used to designate the directions of the x, y, and z axes respectively.
The Cartesian unit vectors have a dimensionless magnitude of 1 and a sense that is given by either a plus or minus sign to show whether they are pointing along the positive or negative x, y or z axes.
11
ES 1022y Engineering Statics
Force Vectors
Two-Dimensional or Coplanar Forces Now consider a two-dimensional, or co-planar, force vector F, as shown.
Resolving the force into components acting along the x and y axes allows us to write the force F as
The magnitudes of each component of F are represented by the positive scalars Fx and Fy. These are often referred to as the rectangular components of F. If we define θ as the angle between the line of action of the force F and the positive x axis, then we can write
Force Addition Using Components Consider three coplanar forces (forces lying in the same plane) F1, F2, and F3.
12
ES 1022y Engineering Statics
Force Vectors
These can be represented using Cartesian vector notation as
The resultant force vector is then given by
In the general case we can write
It is important to remember to take sign conventions into account. Components along positive coordinate axes have positive values and vice versa.
13
ES 1022y Engineering Statics
Force Vectors
The magnitude of the resultant force vector is given by
while the angle is given by
Three-Dimensional Force Vectors In many situations we need to solve problems in three dimensions, rather than the two that we have considered so far. To do this we use a three-dimensional, rectangular Cartesian coordinate system that is said to be right handed. Consider a vector A in 3-D space.
.
The vector can be represented as
with a magnitude given by
14
ES 1022y Engineering Statics
Force Vectors
The orientation of A is defined by the coordinate direction angles α, β, and γ. These are measured between the tail of A and the positive x, y, and z axes respectively, and will always be between 0o and 180o.
The angles are defined by the direction cosines
Recall that in Cartesian vector form A can be written as
The unit vector in the direction of A is then
15
ES 1022y Engineering Statics
Force Vectors
This implies that the unit vector can be rewritten in terms of the direction cosines
Recall that the magnitude of a vector is obtained as
Therefore the magnitude of the unit vector is given by
Since
we can also write A in terms of the coordinate direction angles
If one of the coordinate direction angles is missing, we can always work out what it is by using the equation
and rearranging it to find the cosine squared of the missing angle. The only catch is that the cosine itself can either be positive (angle is less than 90o), or negative (angle is greater than 90o). To determine which is the correct choice we need to determine whether the component of the force acting along the axis associated with the missing angle is acting in the positive direction (cosine is positive and the angle is less than 90o), or the negative direction (cosine is negative and the angle is greater than 90o),
16
ES 1022y Engineering Statics
Force Vectors
As an alternative to using coordinate direction angles, the direction of a force A can also be described using two angles θ and , such as those shown above. In this example the components of A can be found by first resolving the force into vertical and horizontal components using the angle in the vertical plane, which yields
The angle θ lying in the horizontal plane can then be used to resolve the horizontal component of the resolved force into components acting along the x and y axes giving
Finally, force of A can be written in component form as
where the angles θ and are related to the coordinate direction angles α, β, and γ by the following expressions
The above equations should not be memorized; instead it is important to understand how trigonometry was used to determine the components.
17
ES 1022y Engineering Statics
Force Vectors
Example Problem
Determine the magnitude and coordinate direction angles of the resultant force acting on the bracket.
18
ES 1022y Engineering Statics
Force Vectors
19
ES 1022y Engineering Statics
Force Vectors
Example Problem
If the resultant force acting on the bracket is directed along the positive y axis, determine the magnitude of the resultant force and the coordinate direction angles of F so that β < 90o.
20
ES 1022y Engineering Statics
Force Vectors
21
ES 1022y Engineering Statics
Force Vectors
Position Vectors A position vector is a vector that locates a point in space relative to another point.
Consider a position vector extending from the origin O to a point P (x, y, z) in space. The position vector r of the point P relative to the origin O is then given by
In the more general case given a point A (xA, yA, zA) representing the tail of a vector and another point B (xB, yB, zB) representing the head of the vector, the position vector rAB is given by
22
ES 1022y Engineering Statics
Force Vectors
Force Vector Directed Along a Line If a force is directed along a line and the position of two points along the line is known the force vector can be represented in Cartesian coordinates as follows:
This is very useful if we are trying to represent a force vector acting along a cable with known starting and ending coordinates, remembering that the force in the cable will always be acting in tension (away from the point that the force is acting at).
23
ES 1022y Engineering Statics
Force Vectors
Example Problem
The antenna tower is supported by three cables. If the forces of these cables acting on the antenna are FB = 520 N, FC = 680 N, and FD = 560 N, determine the magnitude and coordinate direction angles of the resultant force acting at A.
24
ES 1022y Engineering Statics
Force Vectors
25
ES 1022y Engineering Statics
Force Vectors
Dot Product
The dot product of two vectors A and B, written as A·B, is defined as the product of the magnitudes of A and B and the cosine of the angle θ between their tails. Mathematically it can be written as
Note that the result of the dot product is a scalar and not a vector, with units that are the product of the units associated with vectors A and B. Now consider the dot product of the Cartesian unit vectors i, j, and k.
If we now express vectors A and B in Cartesian form we can write
The dot product has two important applications in statics. Finding the angle between two vectors Given two vectors A and B, the angle θ between the two vectors can be found by
26
ES 1022y Engineering Statics
Force Vectors
Component of a force vector parallel to a particular direction or line
If the direction of the line is specified by the unit vector u, the projection of vector A onto the line is given by
Since the result is a scalar, the vector representation of A|| is given by
The magnitude of the component of vector A perpendicular to the direction or line can be found by using either or the theorem of Pythagoras while the vector form can be found from
27
ES 1022y Engineering Statics
Force Vectors
Example Problem
Determine the projected component of the force FAB = 560 N acting along the cable AC. Express the resultant as a Cartesian vector.
28
ES 1022y Engineering Statics
Force Vectors
29
ES 1022y Engineering Statics
Force Vectors
Example Problem
Determine the angle θ between the two cables attached to the post.
30
ES 1022y Engineering Statics
Force Vectors
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