ENGR-1100 Introduction to Engineering Analysis. Lecture 4

Transcription

1 ENGR-1100 Introduction to Engineering nalysis Lecture 4

2 EQUILIBRIUM OF PRTICLE, THE FREE-BODY DIGRM & COPLNR FORCE SYSTEMS Today s Objectives: Students will be able to : a) Draw a free body diagram (FBD), and, b) pply equations of equilibrium to solve a 2-D problem. In-Class ctivities: pplications What, Why and How of a FBD Equations of Equilibrium nalysis of Spring and Pulleys

3 PPLICTIONS The crane is lifting a load. To decide if the straps holding the load to the crane hook will fail, you need to know the force in the straps. How could you find the forces? Straps

4 PPLICTIONS (continued) For a spool of given weight, how would you find the forces in cables B and C? If designing a spreader bar like this one, you need to know the forces to make sure the rigging doesn t fail.

5 PPLICTIONS (continued) For a given force exerted on the boat s towing pendant, what are the forces in the bridle cables? What size of cable must you use?

6 COPLNR FORCE SYSTEMS (Section 3.3) This is an example of a 2-D or coplanar force system. If the whole assembly is in equilibrium, then particle is also in equilibrium. To determine the tensions in the cables for a given weight of cylinder, you need to learn how to draw a free body diagram and apply the equations of equilibrium.

7 THE WHT, WHY ND HOW OF FREE BODY DIGRM (FBD) Free body diagrams are one of the most important things for you to know how to draw and use for statics and other subjects! What? - It is a drawing that shows all external forces acting on the particle. Why? - It is key to being able to write the equations of equilibrium which are used to solve for the unknowns (usually forces or angles).

8 How? 1. Imagine the particle to be isolated or cut free from its surroundings. 2. Show all the forces that act on the particle. ctive 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. y FBD at F D F B 30 x F C = N (What is this?) Note : Cylinder mass = 40 Kg

9 FBD at F D EQUTIONS OF 2-D EQUILIBRIUM y F B 30 x F C = N FBD at Or, written in a scalar form, F x = 0 and F y = 0 Since particle is in equilibrium, the net force at is zero. So F B + F C + F D = 0 or F = 0 In general, for a particle in equilibrium, F = 0 or F x i + F y j = 0 = 0 i + 0 j (a vector equation) These are two scalar equations of equilibrium (E-of-E). They can be used to solve for up to two unknowns.

10 EQUTIONS OF 2-D EQUILIBRIUM (continued) FBD at F D y F B 30 x F C = N Note : Cylinder mass = 40 Kg Write the scalar E-of-E: + F x = F B cos 30º F D = 0 + F y = F B sin 30º N = 0 Solving the second equation gives: F B = 785 N From the first equation, we get: F D = 680 N

11 SIMPLE SPRINGS Spring Force = spring constant * deformation of spring or F = k * s

12 CBLES ND PULLEYS With a frictionless pulley and cable T 1 = T 2. T 1 T2

13 EXMPLE Given: The box weighs 550 lb and geometry is as shown. Find: The forces in the ropes B and C. Plan: 1. Draw a FBD for point. 2. pply the E-of-E to solve for the forces in ropes B and C.

14 EXMPLE (continued) F B 30 y F C x FBD at point F D = 550 lb pplying the scalar E-of-E at, we get; + F x = F B cos 30 F C (4/5) = 0 + F y = F B sin 30 + F C (3/5) lb = 0 Solving the above equations, we get; F B = 478 lb and F C = 518 lb

15 REDING QUIZ 1) When a particle is in equilibrium, the sum of forces acting on it equals. (Choose the most appropriate answer) ) constant B) positive number C) Zero D) negative number E) n integer 2) For a frictionless pulley and cable, tensions in the cable (T 1 and T 2 ) are related as. ) T 1 > T 2 B) T 1 = T 2 C) T 1 < T 2 T 1 T 2 D) T 1 = T 2 sin

16 CONCEPT QUESTIONS 1000 lb 1000 lb 1000 lb ( ) ( B ) ( C ) 1) ssuming you know the geometry of the ropes, you cannot determine the forces in the cables in which system above? 2) Why? ) 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 lb weight.

17 TTENTION QUIZ 1. Select the correct FBD of particle lb ) 100 lb F C) lb B) D) F1 30 F2 40 F lb F2

18 TTENTION QUIZ 2. Using this FBD of Point C, the sum of forces in the x-direction ( F X ) is. Use a sign convention of +. ) F 2 sin = 0 20 lb C F 1 50 F 2 B) F 2 cos = 0 C) F 2 sin 50 F 1 = 0 D) F 2 cos = 0

19 GROUP PROBLEM SOLVING Given: The mass of lamp is 20 kg and geometry is as shown. Find: The force in each cable. Plan: 1. Draw a FBD for Point D. 2. pply E-of-E at Point D to solve for the unknowns (F CD & F DE ). 3. Knowing F CD, repeat this process at point C.

20 GROUP PROBLEM SOLVING (continued) FBD at point D y F DE F CD D 30 x W = 20 (9.81) N pplying the scalar E-of-E at D, we get; + F y = F DE sin (9.81) = 0 + F x = F DE cos 30 F CD = 0 Solving the above equations, we get: F DE = 392 N and F CD = 340 N

21 GROUP PROBLEM SOLVING (continued) F C FBD at point C y C F CD =340 N x F BC 45 pplying the scalar E-of-E at C, we get; + F x = 340 F BC sin 45 F C (3/5) = 0 + F y = F C (4/5) F BC cos 45 = 0 Solving the above equations, we get; F BC = 275 N and F C = 243 N