3 EQUILIBRIUM System isolation and equilibrium conditions 1

Objectives Students must be able to #1 Course Objective • State the conditions of equilibrium, draw free body diagrams (FBDs), analyse and solve problems involving rigid bodies in equilibrium. Chapter Objectives • Analyse objects (particles and rigid bodies) in equilibriums • Classify problems in equilibrium into SD and SI categories 2

Equilibrium Definition stationary An object is in equilibrium when it is stationary or in steady translation relative to an “inertial reference frame”. moving with Constant Whether object in stationary (moving in steady translation) or not, depends on “reference frame”. velocity really equilibrium? Equilibrium Centrifugal acceleration Earth Newtonian Mechanics “Inertial Reference Frame” - Earth Frame - Central Universe Frame 3

3/1 Introduction A -Equilibrium is the most important subject in statics. A - In statics, we deal primarily with bodies at rest. (i. e. they are in the state of “equilibrium”). -More precisely, when a body is in equilibrium, the wrench resultant of all forces acting on it is zero; i. e. -These requirements are necessary and sufficient conditions for equilibrium; i. e. If is true body in equilibrium If body in equilibrium is true - From the Newton’s second law of motion, a body that moves with constant velocity, rotates with constant angular velocity; i. e. “zero acceleration”, can also be treated as in a state of equilibrium. 4

Equilibrium in 2 D -All physical bodies are inherently 3 D, but many may be treated as 2 D; e. g. when all forces are on the same plane. 3/2 Mechanical system isolation - Before we apply the equilibrium conditions we need to know what force or couple are involved. Isolate body - Draw Free Body Diagram (FBD) FBD is used to isolate body (or bodies / system of bodies) so that force/couple acting on it can be identified. 5

Free Body Diagram (FBD) • FBD is the sketch of the body under consideration that is isolated from all other bodies or surroundings. • The isolation of body clearly separate cause and effects of loads on the body. • A thorough understanding of FBD is most vital for solving problems. 6

body in interest Construction of FBD 1) Pick body/combination of bodies to be isolated 2) Isolating the body. Draw “complete external boundary” of the isolated body 3) Add all forces and moments (including that are applied by the removed surrounding) Free Body Diagram y 4) Indicate a coordinate system 150 cm x 5) Indicate necessary dimensions Most important step is solving problems in mechanics. *** If an FBD is not drawn (when it is needed), you will get no credit ( 0 point ) for the whole problem!!!! *** y 150 cm x 7

Note on drawing FBD Free Body Diagram • Establish the x, y, z axes in any suitable orientation. • Label all the known and unknown force magnitudes and directions on the diagram • The sense of a force having an unknown magnitude can be assumed. Use different colours in diagrams • Body outline • Load (force and couple) • Miscellaneous (dimension, angle, etc. ) y 150 cm x - blue - red - black 8

Equilibrium Solving Procedure • Formulate problems from physical situations. (Simplify problems by making appropriate assumptions) • Draw the free body diagram (FBD) of objects under consideration • State the condition of equilibrium • Substitute variables from the FBD into the equilibrium equations • Substitute the numbers and solve for solutions – Delay substitute numbers – Use appropriate significant figures • Technical judgment and engineering sense – Try to predict the answers – Is the answer reasonable? 9

Equilibrium Free Body Diagram (FBD) • FBD is the sketch of the body under consideration that is isolated from all other bodies or surroundings. • The isolation of body clearly separate cause and effects of loads on the body. • A thorough understanding of FBD is most vital for solving problems. 10

Equilibrium FBD Construction • • Select the body to be isolated Draw boundary of isolated body, excluding supports Indicate a coordinate system by drawing axes Add all applied loads (forces and couples) on the isolated body. • Add all to support reactions (forces and couples) represent the supports that were removed. • Beware of loads or support reactions with specific directions due to physical meanings • Add dimensions and other information that are required in the equilibrium equation 11

Equilibrium Help on FBD Establish the x, y axes in any suitable orientation. n Label all the known and unknown applied load and support reaction magnitudes n Beware of loads or support reactions with specific directions due to physical meanings n Otherwise, directions of unknown loads and support reactions can be assumed. n 12

Equilibrium On FBD Analyses Objective: To find support reactions • Apply the equations of equilibrium • Load components are positive if they are directed along a positive direction, and vice versa • It is possible to assume positive directions for unknown forces and moments. If the solution yields a negative result, the actual load direction is opposite of that shown in the FBD. 13

Contents • Equilibrium of Objects – Particles (2 D & 3 D) – 2 D Rigid Bodies – 3 D Rigid Bodies the heart ( ) of Statics Rigid Bodies • SD and SI Problems Particle Ideal particle can not rotate. (no couple acting on it) 14

Particles FBD construction To construct a complete FBD of a particle • Select the particle to be isolated • Draw the particle as a point • Indicate a coordinate system • Add all active forces/moments (weight, etc. ) • Add all support reactions (e. g. tension in the tangential direction of a cable, tensile and compressive forces in a compressed and stretched springs) 15

Particles Equilibrium Analyses #2 Equations of Equilibrium • Apply the equations of equilibrium • Components are positive if they are directed along a positive axis, and negative if they are directed along a negative axis. • Assume the directions of unknown forces in the positive x, and y axes. If the solution yields a negative result, this indicates the sense of the force is the reverse of that shown on the FBD. 16

Particle Equilibrium Particles Equilibrium in 3 D z y x 17

H-Ex 3 -1#1) The sphere has a mass of 6 kg and is supported as shown. Draw a free-body diagram of the sphere, the cord CE, and the knot at C. 18

How many unknowns? How many Equations? y x Action-reaction pair - use same symbol - opposite direction

Determine the tension in cables AB and AD for equilibrium of the 250 -kg engine shown. FBD of A 20

FBD of A 21

Example Hibbeler Ex 3 -3 #1 2 unknown, 2 Eqs (at this stage) 3 unknown, 2 Eqs (at this stage) If the sack at A has a weight of 20 lb, determine the weight of the sack at B and the force in each cord needed to hold the system in the equilibrium position shown. =? FBD of E 22

Recommended FBD of E FBD of C 23

Example Hibbeler Ex 3 -5 #1 A 90 -N load is suspended from the hook. The load is supported by two cables and a spring having a stiffness k = 500 N/m. Determine the force in the cables and the stretch of the spring for equilibrium. Cable AD lies in the x-y plane and cable AC lies in the x-z plane. 24

3 D particle Equilibrium How many unknowns, how many equations? (no FBD, no score) perfect answer sheet 25

Example Hibbeler Ex 3 -7 #1 3 D particle Equilibrium Determine the force developed in each cable used to support the 40 -k. N (4 tonne) crate shown. 26

Particle Equilibrium Hibbeler Ex 3 -7 #2 27

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Particle Equilibrium Example Hibbeler Ex 3 -7 #4 29

2 D Equilibrium of 2 D Rigid Bodies • Use similar analyses as the particles • Additional consideration – Action forces in supports/constraints – Free-body diagram (FBD) of 2 D rigid bodies – Equilibrium equations (scalar form) for rigid bodies – Two-force and three-force members 31

Force Reaction (2 D) To write an FBD, first, you will need to know what kind of force we will get when eliminating the environment/surrounding. 1) Flexible cable, belt, chain, or rope T always away from the body tangent to the cable T 32

2) Smooth surfaces N only this direction N 3) Rough surfaces Not always this direction - Contact force at contact point normal to the surface/contact plane - always compressive -A rough surface can produce a tangential force (F, friction) as well as a normal force (N) F R N only this direction - direction of F depend on situations (chapter 6) 33

4) Roller supports N - Roller, rocker, or ball transmits a compressive force normal to the supporting surface N 5) Freely sliding guide not always this direction only this direction The vector N may be up or down depend on problem. If not known, you may assume any of the two. After further calculation, if N is +, correct sense was assumed. If negative, N goes the other way. N M existence due to its bending resistance 34

6) Pin connection Pin not free to turn Pin free to turn Rx not always this direction Rx Ry M Ry As a general rule, if a support prevents translation of a body in a given direction, then a force is developed on the body in the opposite direction. Similarly, if rotation is prevented, a couple is exerted on the body. 7) Built-in or fixed support F M not always this direction V A A Weld A 35

8) Gravitational attraction m G W=mg Resultant of the gravitational attraction is the weight W = mg and act toward center of the earth passing through the center mass G 9) Spring action x F x is positive x is negative F F Normal distance For linear springs , F = kx 36

Equilibrium You may assume either case. The sign will indicate its sense of direction later. 37

2 D Equilibrium construction of a FBD To construct a complete FBD for a 2 D rigid bodies • determine which body is to be isolated • draw external boundary of isolated body • indicate a coordinate system (axes) • add all loads (forces and couples, be they applied or support) No FBDs Cannot apply equilibrium conditions NO SCORES 38

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