MAE206-L06-Ch3.3. Equilibrium of Particles in 3D

MAE 206 Engineering Statics Prof. Mary Zadeh Assistant Teaching Professor in Mechanical and Aerospace Engineering Department Fall 2023 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 6 – Prof. Zadeh Lecture 6 Equilibrium of Particles Ch 3.3. Equilibrium of Particles in 3D

Ch. 3.3 Page: 2 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 6 – Prof. Zadeh 1 0 . M o m e n t s o f I n e r t i a 9 . F r i c t i o n 7 . C e n t r o i d s & D i s t r i b u t e d F o r c e S y s t e m s 8 . I n t e r n a l F o r c e s 6 . S t r u c t u r a l A n a l y s i s a n d M a c h i n e s 5 . E q u i l i b r i u m o f b o d i e s 4 . M o m e n t o f a F o r c e 3 . E q u i l i b r i u m o f P a r t i c l e s 2 . V e c t o r s : F o r c e a n d P o s i t i o n 1 . I n t r o d u c t i o n t o S t a t i c s MAE 206 Engineering Statics

Ch. 3.3 Page: 4 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 6 – Prof. Zadeh Review • Problem-Solving Approach ▪ Road Map: Identify givens/data and unknowns and create a solution strategy. ▪ Modeling: Identify the system and make reasonable assumptions leading to the development of a FBD (or possibly multiple FBDs). ▪ Sketch the particle. Isolate it from its environment. Draw all the forces acting on it. Select a coordinate system , show dimensions and orientations of all forces. ▪ Governing Equations: Write the equations of equilibrium σ 𝑭 = 𝟎 , and as needed, additional eqns for other physical phenomena pertinent to problem. ▪ Computation: Solve the system of governing equations. ▪ Discussion and Verification: The results are critically examined and interpreted . • Cables ▪ Support only tensile forces • Bars ▪ Support both tensile and compressive forces • Pulleys

Ch. 3.3 Page: 5 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 6 – Prof. Zadeh Review of Chapter 2 • force vector Ԧ 𝐹 lies along the line of action of a position vector Ԧ 𝑟 : Ԧ 𝐹 = 𝐹 ො𝑢 𝑟 = 𝐹 Ԧ 𝑟 Ԧ 𝑟 • Component of a Vector in a Particular Direction • Vector component parallel to Ԧ 𝑟 : 𝐹 ∥ = Ԧ 𝐹 ⋅ Ԧ 𝑟 Ԧ 𝑟 • Vector component perpendicular to Ԧ 𝑟 : 𝐹 ⊥ = 𝐹 2 − 𝐹 ∥ 2 as Ԧ 𝐹 = Ԧ 𝐹 ∥ + Ԧ 𝐹 ⊥

Ch. 3.3 Page: 7 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 6 – Prof. Zadeh Solution of Algebraic Equations • Analysis of equilibrium of one particle in three dimensions will typically produce three simultaneous linear algebraic equations with three unknowns . • Efficient techniques for solving systems of simultaneous algebraic equation: ▪ Scientific calculators, ▪ Programs such as Mathematica, Maple, Mathcad, EES, MATLAB.

Ch. 3.3 Page: 8 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 6 – Prof. Zadeh Reactions • If a support prevents motion of a particle in a certain direction, it can do so only by producing a reaction force in that direction.

Ch. 3.3 Page: 10 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 6 – Prof. Zadeh Example 3.7 Summing Forces in a Direction Other Than x, y, or z Bar AB is straight and fixed in space. Spring CD has 3 𝑁/𝑚𝑚 stiffness and 200 𝑚𝑚 unstretched length. If there is no friction between collar C and bar AB, determine: (a) The weight W of the collar that produces the equilibrium configuration shown. (b)The reaction between the collar and bar AB.

Ch. 3.3 Page: 11 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 6 – Prof. Zadeh Example 3.7 Solution