Engineering Mechanics : STATICS Lecture #03 By, Noraniah Binti Kassim Hairul Mubarak b. Hassim University Tun Hussein Onn Malaysia (UTHM),
DAJ 21003 ( Statics & Dynamics)
EQUILIBRIUM OF A PARTICLE IN 2-D Today’s Objectives: Students will be able to : a) Draw a free body diagram (FBD), and, b) Apply equations of equilibrium to solve a 2-D problem. Learning Topics: What, why and how of a FBD Equations of equilibrium Analysis of spring and pulleys DAJ 21003 ( Statics & Dynamics)
DAJ 21003 ( Statics & Dynamics) READING QUIZ 1) When a particle is in equilibrium, the sum of forces acting on it equals ___ . (Choose the most appropriate answer) A) a constant B) a positive number C) zero D) a negative number E) an integer. 2) For a frictionless pulley and cable, tensions in the cable (T1 and T2) are related as _____ . A) T1 > T2 B) T1 = T2 C) T1 < T2 D) T1 = T2 sin Answers: 1. C 2. B DAJ 21003 ( Statics & Dynamics)
DAJ 21003 ( Statics & Dynamics) APPLICATIONS For a spool of given weight, what are the forces in cables AB and AC ? DAJ 21003 ( Statics & Dynamics)
APPLICATIONS (continued) For a given cable strength, what is the maximum weight that can be lifted ? DAJ 21003 ( Statics & Dynamics)
EQUILIBRIUM OF PARTICLE IN 2-D (Section 3.3) This is an example of a 2-D or coplanar force system. If the whole assembly is in equilibrium, then particle A is also in equilibrium. To determine the tensions in the cables for a given weight of the engine, we need to learn how to draw a free body diagram and apply equations of equilibrium. DAJ 21003 ( Statics & Dynamics)
THE WHAT, WHY AND HOW OF A FREE BODY DIAGRAM (FBD) Free Body Diagrams are one of the most important things for you to know how to draw and use. What ? - It is a drawing that shows all external forces acting on the particle. Why ? - It helps you write the equations of equilibrium used to solve for the unknowns (usually forces or angles). DAJ 21003 ( Statics & Dynamics)
DAJ 21003 ( Statics & Dynamics) How ? 1. Imagine the particle to be isolated or cut free from its surroundings. 2. Show all the forces that act on the particle. Active forces: They want to move the particle. Reactive forces: They tend to resist the motion. 3. Identify each force and show all known magnitudes and directions. Show all unknown magnitudes and / or directions as variables . Note : Engine mass = 250 Kg FBD at A DAJ 21003 ( Statics & Dynamics)
EQUATIONS OF 2-D EQUILIBRIUM Since particle A is in equilibrium, the net force at A is zero. So FAB + FAD + FAC = 0 or F = 0 In general, for a particle in equilibrium, F = 0 or Fx i + Fy j = 0 = 0 i + 0 j (A vector equation) Or, written in a scalar form, Fx = 0 and Fy = 0 These are two scalar equations of equilibrium (EofE). They can be used to solve for up to two unknowns. DAJ 21003 ( Statics & Dynamics)
DAJ 21003 ( Statics & Dynamics) EXAMPLE Note : Engine mass = 250 Kg FBD at A Write the scalar EofE: + Fx = TB cos 30º – TD = 0 + Fy = TB sin 30º – 2.452 kN = 0 Solving the second equation gives: TB = 4.90 kN From the first equation, we get: TD = 4.25 kN DAJ 21003 ( Statics & Dynamics)
SPRINGS, CABLES, AND PULLEYS Spring Force = spring constant * deformation, or F = k * S With a frictionless pulley, T1 = T2. DAJ 21003 ( Statics & Dynamics)
DAJ 21003 ( Statics & Dynamics) EXAMPLE Given: Sack A weighs 20 N. and geometry is as shown. Find: Forces in the cables and weight of sack B. Plan: 1. Draw a FBD for Point E. 2. Apply EofE at Point E to solve for the unknowns (TEG & TEC). 3. Repeat this process at C. You may ask the students to give a plan You may explain why analyze at E, first, and C, later DAJ 21003 ( Statics & Dynamics)
DAJ 21003 ( Statics & Dynamics) EXAMPLE (continued) A FBD at E should look like the one to the left. Note the assumed directions for the two cable tensions. The scalar EofE are: + Fx = TEG sin 30º – TEC cos 45º = 0 + Fy = TEG cos 30º – TEC sin 45º – 20 N = 0 You may ask students to give equations of equilibrium.students hand-outs should be without the equations. Solving these two simultaneous equations for the two unknowns yields: TEC = 38.6 N TEG = 54.6 N DAJ 21003 ( Statics & Dynamics)
DAJ 21003 ( Statics & Dynamics) EXAMPLE (continued) Now move on to ring C. A FBD for C should look like the one to the left. The scalar EofE are: Fx = 38.64 cos 45 – (4/5) TCD = 0 Fy = (3/5) TCD + 38.64 sin 45 – WB = 0 You may ask the students to give equations of equilibrium.Students hand-outs should be without the equation. You may also explain about the direction of 38.6 lb force (see fig. 3.7 d, page 88) Solving the first equation and then the second yields TCD = 34.2 N and WB = 47.8 N . DAJ 21003 ( Statics & Dynamics)
DAJ 21003 ( Statics & Dynamics) CONCEPT QUESTIONS 1000 N 1000 N 1000 N ( A ) ( B ) ( C ) 1) Assuming you know the geometry of the ropes, you cannot determine the forces in the cables in which system above? 2) Why? A) The weight is too heavy. B) The cables are too thin. C) There are more unknowns than equations. D) There are too few cables for a 1000 N weight. Answers: 1. C 2. C You may ask the students to record their answers, first, without group discussion and later after the group discussion with their neighbors. Again, like after reading quiz, they should verbally indicate the answers. You can comment on the answers, emphasizing statically indeterminate condition. DAJ 21003 ( Statics & Dynamics)
IN CLASS TUTORIAL (GROUP PROBLEM SOLVING) Given: The car is towed at constant speed by the 600 N force and the angle is 25°. Find: The forces in the ropes AB and AC. Plan: 1. Draw a FBD for point A. 2. Apply the EofE to solve for the forces in ropes AB and AC. DAJ 21003 ( Statics & Dynamics)
GROUP PROBLEM SOLVING (continued) 30° 25° 600 N FAB FAC A FBD at point A Applying the scalar EofE at A, we get; + Fx = FAC cos 30° – FAB cos 25° = 0 + Fy = -FAC sin 30° – FAB sin 25° + 600 = 0 Solving the above equations, we get; FAB = 634 N FAC = 664 N DAJ 21003 ( Statics & Dynamics)
DAJ 21003 ( Statics & Dynamics) ATTENTION QUIZ 1. Select the correct FBD of particle A. A 30 40 100 N F1 F2 A A) B) 30 40° Answers: 1. D 100 N A F2 F F1 D) C) 30° 40° 30° A A 100 N 100 N DAJ 21003 ( Statics & Dynamics)
DAJ 21003 ( Statics & Dynamics) ATTENTION QUIZ 2. Using this FBD of Point C, the sum of forces in the x-direction ( FX) is ___ . Use a sign convention of + . A) F2 sin 50° – 20 = 0 B) F2 cos 50° – 20 = 0 C) F2 sin 50° – F1 = 0 D) F2 cos 50° + 20 = 0 20 N 50° C F1 Answers: 2. B DAJ 21003 ( Statics & Dynamics)
DAJ 21003 ( Statics & Dynamics) HOMEWORK TUTORIAL Q1 (2.44) : Knowing that α = 25°, determine the tension in cable AC, in rope BC. DAJ 21003 ( Statics & Dynamics)
HOMEWORK TUTORIAL (continued) Q2 (2.46) : Two cables are tied together at C and are loaded as shown. Knowing that α = 30°, determine the tension (a) in cable AC, (b) in cable BC. DAJ 21003 ( Statics & Dynamics)
HOMEWORK TUTORIAL (continued) Q3 (2.51) : Two forces P and Q are applied as shown to an aircraft connection. Knowing that the connection is in equilibrium and the P = 1.8kN and Q = 2.3 kN, determine the magnitudes of the forces exerted on the rods A and B. DAJ 21003 ( Statics & Dynamics)
HOMEWORK TUTORIAL (continued) Q4 (2.67) : A 280-kg crate is supported by several rope-and-pulley arrangements as shown. Determine for each arrangement the tension in the rope. (Hint: The tension in the rope is the same on each side of a simple pulley.) T T T T T (a) (b) (c) (d) (e) DAJ 21003 ( Statics & Dynamics)
HOMEWORK TUTORIAL (continued) Q5 (3-13) : Determine the stretch in each spring for equilibrium of the block of mass M. The springs are shown in the equilibrium position. Given: M = 2kg a = 3m b = 3m c = 4m kAB=30Nm kAC = 20Nm kAD = 40Nm g = 9.81ms² DAJ 21003 ( Statics & Dynamics)
HOMEWORK TUTORIAL (continued) Q6 (3-17) : Determine the force in each cable and the force F needed to hold the lamp of mass M in the position shown. Hint: First analyze the equilibrium at B; then, using the result for the force in BC, analyze the equilibrium at C. Given: M := 4kg θ1 := 30° θ2 := 60° θ3 := 30° DAJ 21003 ( Statics & Dynamics)
DAJ 21003 ( Statics & Dynamics) End of the Lecture DAJ 21003 ( Statics & Dynamics)