CE297-A09-Ch2 Page 1 Wednesday, August 26, 2009 4:18 AM Chapter 2: Statics of Particles 2.1-2.3 orces as Vectors & Resultants orces are drawn as directed arrows. The length of the arrow represents the magnitude of the force and the arrow shows its direction. orces on rigid bodies further have a line of action. orces (and in general all vectors) follow the parallelogram law of vector addition. In fact, vectors are defined as quantities that follow the parallelogram law. Q R R = P + Q P Vector addition is represented by the same symbol + The meaning of plus will be clear from the context it is used in. Note: Vector addition is independent of any chosen coordinate system. Two vectors are equal if they have the same magnitude and direction. ' The negative of a vector is simply - denoted by arrow of the same size in the opposite direction. + (-) = 0 - -
CE297-A09-Ch2 Page 2 Wednesday, August 26, 2009 11:05 AM 2.4-2.5 Addition of Vectors, orce resultants Vector Addition Parallelogram law: Commutative property P + Q = Q + P = R Vector Addition Triangle law (tip-to-tail): Derives from the parallelogram law. Subtraction of a vector from another vector: (Addition of the negative vector) P - Q = P + (-Q) Alternatively Addition of multiple vectors Associative property P + Q + S = (P + Q) + S = P + (Q + S) = S + Q + P
CE297-A09-Ch2 Page 3 Alternatively Using tip-to-tail rule (Polygon Law): Product of a scalar & a vector "Scales" the length of the vector. Compare 1, 2 and 3. scalar vector Similarly
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CE297-A09-Ch2 Page 5 riday, August 28, 2009 7:02 AM Examples Read Example 2.1 in book. Exercise 2.1 P = 75 N Q = 125 N Determine the resultant using (1) Parallelogram Law, (2) Triangle Rule, (3) Trigonometry. A A Using trigonometry A B C Using the cosine Law :- solving:
Now using Sine Law: CE297-A09-Ch2 Page 6
CE297-A09-Ch2 Page 7 riday, August 28, 2009 4:08 AM 2.6 Resolving orces into Components Reverse process of vector addition. Split a force into two (or more) components. Given, ind P and Q? = P + Q Many Possibilities Special cases: One component (say P) is known: The line of action of both components is known
CE297-A09-Ch2 Page 8 riday, August 28, 2009 10:14 AM Example 2.2 in book: 5000 lb (1) If α=45, then find T1 and T2 (2) The value of α for which T2 is minimum. (1) (2) In class exercise.
CE297-A09-Ch2 Page 9 Thursday, August 27, 2009 5:19 PM 2.7 Rectangular Components of Vectors; Unit vectors or ease in mathematical manipulation, forces (and vectors) can be resolved into rectangular components along predefined x, y (and z) directions. y y y' y' x' x x x' = x + y = x' + y' One can choose any coordinate system [O, i, j, k] and resolve forces and vectors along these directions. j j' i' i, j, (and k) are unit vectors. (magnitude =1) O i O' Note: i' = cosθ i + sinθ j j' = - sinθ i + cosθ j i' cosθ sinθ i j' - sinθ cosθ j Useful in converting one coordinate system to another. Using the unit vectors: y y x = x i y = y j x = x + y = x i + y j x Vector Unit Vector (Direction) Scalar (Magnitude) x = cosθ y = sinθ Examples Read examples 1, 2 & 3 in section 2.7 of the book.
CE297-A09-Ch2 Page 10 Exercise 2.21 Determine the x, y components of the 3 forces. (i) Note: Similarly solve for the other two forces. 2. 8 Addition of forces in x-y components The vector sum of two or more forces can be obtained by Resolving each force into x-y components Simply adding the individual components
CE297-A09-Ch2 Page 11 where Example Read example 2.3 in book. Exercise 2.36 Tension in BC = 725 N. Determine the resultant of the 3 forces at B. Note:
CE297-A09-Ch2 Page 12 Thus the resultant is: 2.9-2.11 Equilibrium of a Particle; ree Body Diagrams If the resultant of all the forces acting on a particle is zero, the particle is said to be in equilibrium. R = Σ = 0 Σ x = 0 ; Σ y = 0 Recall, Newton's 1st Law. Choose the particle judiciously. REE BODY DIAGRAMS A diagram showing all the forces acting on a particle or an object.
CE297-A09-Ch2 Page 13 In 2 dimensions, we have two equations of equilibrium, thus the maximum number of unknowns we can solve for is two. Examples: Unloading a truck. A Read Examples 2.4, 2.5 and 2.6 in book. Exercise 2.46: Determine T AC and T BC 586.3604 2188.3
CE297-A09-Ch2 Page 14 Tuesday, September 01, 2009 10:44 PM Pulleys The tension on both sides of a frictionless pulley is the same (under Equilibrium). T T T T T T Exercise 2.65 ind P for equilibrium. We wave to consider the free body diagrams of all the "particles" in the system here.
CE297-A09-Ch2 Page 15 Monday, August 31, 2009 3:24 AM 2.12-2.14 orces (and Vectors) in 3-Dimensional Space Unit vectors in space. scalar = x i + y j + z k Unit vector A force in 3-dimensional space can be represented with the angles φ and θ that it makes with the x-axis and the vertical axis respectively. Vector Note:
CE297-A09-Ch2 Page 16 Example: Consider the force shown: Direction Cosines Example: or the example shown above in figure 1:
CE297-A09-Ch2 Page 17 General 3-dimensional Unit Vector A general 3-D unit vector can be used to represent the line of action of a 3-D force. λ = cos θ x i + cos θ y j + cos θ Z k = λ λ Example or the example in figure 1: λ = 0.354 i + 0.866 j + 0.354 k = 10 λ N Addition of forces (vectors) in 3-D space Simply add the x, y, and z components. R = Σ 2.15 Equilibrium of Particles in 3D Space R = Σ = 0 Σ x = 0 ; Σ y = 0 ; Σ z = 0 Note: 3 Independent equations. Thus 3 unknowns can be solved for. Example 2.7 in book
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CE297-A09-Ch2 Page 19 riday, September 04, 2009 9:47 AM Example ree body diagram of point O:
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CE297-A09-Ch2 Page 21 riday, September 04, 2009 12:00 PM In Class Assignment 2: Side View: Top View: ind: (i) Is the pole OD in "tension" or "compression". (ii) Magnitude of the force in the pole OD. Consider point O: Choose x, y, z such that "y" is vertical; and 1h is in the (-) "x" direction. Note: This is an arbitrary choice. You may choose a different x, y, z, and the result will be the same. Resolving the 3 forces into "vertical" and "horizontal" components, rather than x, y, z. Each of the 3 triangles will be "similar".
CE297-A09-Ch2 Page 22 On the x-z plane: ree body diagram of the pole: