Force Vectors Lecture

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Engineering Science
Engineering Science 1022A/B/Y
Craig Miller

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 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 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 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 F xand F y 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 Component s Consider three coplanar forces (forces lying in the same plane) F , F 1 an2 F . 3 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 o o always be between 0 and 180 . 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
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