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
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 2 2007 S. Kenny, Ph.D., P.Eng.
Lecture 11 Objectives t further examine and apply Chapter 3 bjectives in 2D and 3D space 3 2007 S. Kenny, Ph.D., P.Eng.
Nte n ree Bdy Diagram rce Sense and Slutin Negative sign indicates the frce sense is ppsite that shwn n the BD + y 0 2 1 1 2 0 2 1 mg 2 1 mg 4 2007 S. Kenny, Ph.D., P.Eng.
Omit Ch.3 Spring Prblems 5 2007 S. Kenny, Ph.D., P.Eng.
Example 11-01 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. 6 2007 S. Kenny, Ph.D., P.Eng.
Example 11-01 (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 7 2007 S. Kenny, Ph.D., P.Eng. 45 AH H W mg Pint H 1 Equatin 2 Unknwns but. Newtn s 3 rd Law
Example 11-01 (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 8 2007 S. Kenny, Ph.D., P.Eng.
Example 11-01 (cnt.) BD at Pint A + 0 x cs 60 AB AC cs 45 AB cs 45 + AC cs 60 0.7071 AB 0.7071AC 0.5176N AC 0 y sin 45 + sin60 0 + 0 0 AC AB AC AH.7071AC sin 45 + sin60 W 0 AC.5AC + 0.866 1 AC 0.7321N 0 0 AC AB A 45 60 AH W 1N 9 2007 S. Kenny, Ph.D., P.Eng.
Example 11-01 (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 30 0.5176N cs 45 0.268N y sin 45 + sin 30 0 + 0 BA BE 0.5176N sin 45 + BE sin 30 BE 0.7320 N 0 BE 30 B 45 BC BA AB 0.5176N 10 2007 S. Kenny, Ph.D., P.Eng.
Example 11-01 (cnt.) BD at Pint C + 0 x CD csθ CA cs 60 CB 0 CD csθ 0.7321N cs 60 + 0.268N 0.6341N y sin60 + sinθ 0 + 0 CD sinθ 0.7321N sin 60 θ y x CD CD sinθ csθ 0.6340N tan 1 0.6341N CA CD 0.6340N 0.6341N 45 0.6340N CB BC 0.268N C 11 2007 S. Kenny, Ph.D., P.Eng. 0.6340N CD sinθ 60 0.8966N CD CA AC 0.7321N θ
Example 11-01 (cnt.) Crd rces Analysis summary unit frce AB AC BE BC CD AH 1 0.5176 0.7321 N 0.7320 0.268 0.8966 Maximum frce 200 N 200 AB AC BE BC CD AH 1 0.5176 0.7321 N 0.7320 0.268 0.8966 AH AB AC BE BC CD 200 104 146 N 146 53.6 179 12 2007 S. Kenny, Ph.D., P.Eng.
Example 11-01 (cnt.) Use f Vectr Algebra in Mathematical Sftware t Slve Mechanics Prblems Mathcad www.mathcad.cm Engineering calculatins This discussin n the use f Mathcad is just fr knwledge It is nt part f any curse requirement 13 2007 S. Kenny, Ph.D., P.Eng.
Example 11-01 (cnt.) Mathcad Slutin Set-up equilibrium equatins 14 2007 S. Kenny, Ph.D., P.Eng.
Example 11-01 (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 15 2007 S. Kenny, Ph.D., P.Eng.
Example 11-01 (cnt.) Mathcad Slutin Slve system f equatins 16 2007 S. Kenny, Ph.D., P.Eng.
Particle Equilibrium in 3D Newtn s 1 st Law r r r r 1 + 2 + 3 0 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 0 17 2007 S. Kenny, Ph.D., P.Eng.
Cmprehensin Quiz 11-01 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 18 2007 S. Kenny, Ph.D., P.Eng.
Cmprehensin Quiz 11-02 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 19 2007 S. Kenny, Ph.D., P.Eng.
Cmprehensin Quiz 11-03 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 4 4 4 4 { 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 20 2007 S. Kenny, Ph.D., P.Eng.
Classificatin f Textbk Prblems Hibbeler (2007) 21 2007 S. Kenny, Ph.D., P.Eng.
References Hibbeler (2007) http://wps.prenhall.cm/esm_hibbeler_eng mech_1 22 2007 S. Kenny, Ph.D., P.Eng.