ENGI 1313 Mechanics I
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1 ENGI 1313 Mechanics I Lecture 11: 2D and 3D Particle Equilibrium Shawn Kenny, Ph.D., P.Eng. Assistant Prfessr aculty f Engineering and Applied Science Memrial University f Newfundland spkenny@engr.mun.ca
2 Chapter 3 Objectives t intrduce the cncept f the free-bdy diagram fr a particle. t shw hw t slve particle equilibrium prblems using the equatins f equilibrium S. Kenny, Ph.D., P.Eng.
3 Lecture 11 Objectives t further examine and apply Chapter 3 bjectives in 2D and 3D space S. Kenny, Ph.D., P.Eng.
4 Nte n ree Bdy Diagram rce Sense and Slutin Negative sign indicates the frce sense is ppsite that shwn n the BD + y mg 2 1 mg S. Kenny, Ph.D., P.Eng.
5 Omit Ch.3 Spring Prblems S. Kenny, Ph.D., P.Eng.
6 Example Each crd can sustain a maximum tensin f 200 N Determine the largest weight f the sack that can be supprted. Als, determine θ f crd DC fr equilibrium S. Kenny, Ph.D., P.Eng.
7 Example (cnt.) Where t Start? + x 0 + y 0 BE 30 B 45 BC CB 60 CD BA CA C θ Pint B 2 Equatins 3 Unknwns Pint C 2 Equatins 3 Unknwns Pint A 2 Equatins 3 Unknwns AB A 60 AH AC S. Kenny, Ph.D., P.Eng. 45 AH H W mg Pint H 1 Equatin 2 Unknwns but. Newtn s 3 rd Law
8 Example (cnt.) BD at Pint H What Crd Will Have the Maximum Tensin? Educated guess Experience Theretical apprach Assume W 1N Maximum crd tensin 200 N + y 0 AH H W mg AH W m g S. Kenny, Ph.D., P.Eng.
9 Example (cnt.) BD at Pint A + 0 x cs 60 AB AC cs 45 AB cs 45 + AC cs AB AC N AC 0 y sin 45 + sin AC AB AC AH.7071AC sin 45 + sin60 W 0 AC.5AC AC N 0 0 AC AB A AH W 1N S. Kenny, Ph.D., P.Eng.
10 Example (cnt.) BD at Pint B + 0 x BC + BA cs 45 BE cs 30 0 BC + BA cs 45 BE cs 30 0 BC 0.732N cs N cs N y sin 45 + sin BA BE N sin 45 + BE sin 30 BE N 0 BE 30 B 45 BC BA AB N S. Kenny, Ph.D., P.Eng.
11 Example (cnt.) BD at Pint C + 0 x CD csθ CA cs 60 CB 0 CD csθ N cs N N y sin60 + sinθ CD sinθ N sin 60 θ y x CD CD sinθ csθ N tan N CA CD N N N CB BC 0.268N C S. Kenny, Ph.D., P.Eng N CD sinθ N CD CA AC N θ
12 Example (cnt.) Crd rces Analysis summary unit frce AB AC BE BC CD AH N Maximum frce 200 N 200 AB AC BE BC CD AH N AH AB AC BE BC CD N S. Kenny, Ph.D., P.Eng.
13 Example (cnt.) Use f Vectr Algebra in Mathematical Sftware t Slve Mechanics Prblems Mathcad Engineering calculatins This discussin n the use f Mathcad is just fr knwledge It is nt part f any curse requirement S. Kenny, Ph.D., P.Eng.
14 Example (cnt.) Mathcad Slutin Set-up equilibrium equatins S. Kenny, Ph.D., P.Eng.
15 Example (cnt.) Mathcad Slutin Uses a cmmand ind t slves a system f linear equatins This system f linear equatins is based n the BD analysis that defines the equilibrium equatins (Σ x and Σ y ) The ind cmmand functin requires an initial guess r estimate f the frces and angle (θ) t start the mathematical search f the slutin S. Kenny, Ph.D., P.Eng.
16 Example (cnt.) Mathcad Slutin Slve system f equatins S. Kenny, Ph.D., P.Eng.
17 Particle Equilibrium in 3D Newtn s 1 st Law r r r r Scalar cmpnents 0 Cartesian Vectr r r r x î + y ĵ + z kˆ 0 3 Equatins Slve fr at mst 3 unknwns + x 0 + y 0 + z S. Kenny, Ph.D., P.Eng.
18 Cmprehensin Quiz In 3-D, the directin f a frce is knwn but nt the frce magnitude, hw many unknwns crrespnding t that frce remain? A) One B) Tw C) Three D) ur Answer: A Hint : r r û S. Kenny, Ph.D., P.Eng.
19 Cmprehensin Quiz In 3-D, when yu dn t knw either the directin r magnitude f a frce, hw many unknwns d yu have crrespnding t that frce? A) One B) Tw C) Three D) ur Answer: C Hint : r r û û csα î + cs β ĵ + cs γ kˆ S. Kenny, Ph.D., P.Eng.
20 Cmprehensin Quiz ur frces act at pint A and the system is in equilibrium. Select the crrect frce vectr 4 t balance the system. z A) B ) C ) D ) r r r r Answer: D { 20 î + 10 ĵ 10 kˆ } { 10 î 20 ĵ 10 kˆ } { + 20 î 10 ĵ 10 kˆ } nne f the abve N N N x A 3 10 N 1 20 N 2 10 N y S. Kenny, Ph.D., P.Eng.
21 Classificatin f Textbk Prblems Hibbeler (2007) S. Kenny, Ph.D., P.Eng.
22 References Hibbeler (2007) mech_ S. Kenny, Ph.D., P.Eng.
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