ENGI 11 Mechancs I Lecture 40: Center of Gravty, Center of Mass and Geometrc Centrod Shan Kenny, Ph.D., P.Eng. ssstant Professor Faculty of Engneerng and ppled Scence Memoral Unversty of Nefoundland spkenny@engr.mun.ca
Materal Coverage for Fnal Exam Introducton (Ch.1: Sectons 1.1 1.6) Force Vectors (Ch.2: Sectons 2.1 2.9) Partcle Equlbrum (Ch.: Sectons.1.4) Force System Resultants (Ch.4: Sectons 4.1 4.10) Omt Wrench (p.174) Rgd Body Equlbrum (Ch.5: Sectons 5.1 5.7) Structural nalyss (Ch.6: Sectons 6.1 6.4 & 6.6) Frcton (Ch.8: Sectons 8.1 8.) Center of Gravty and Centrod (Ch.9: Sectons 9.1 9.) Ignore problems nvolvng closed-form ntegraton Smple shapes such as square, rectangle, trangle and crcle 2 2007 S. Kenny, Ph.D., P.Eng.
Lecture 40 Objectve to understand the concepts of center of gravty, center of mass, and geometrc centrod to be able to determne the locaton of these ponts for a system of partcles or a body 2007 S. Kenny, Ph.D., P.Eng.
Center of Gravty Pont locatng the equvalent resultant eght of a system of partcles or body Example: Sold Blocks re both fnal confguratons stable? 5 5 2 2 1 1 W R W R 4 2007 S. Kenny, Ph.D., P.Eng.
Center of Gravty (cont.) Resultant Weght L/2 W R 1K4 Coordnates 4 x W zw R R z~ 1K4 1K4 z ~ z 1 1 x G 2 Key Property M G ( x ) 0 1K4 G x ~ 1 x W R 5 2007 S. Kenny, Ph.D., P.Eng.
Center of Gravty (cont.) Generalzed Formulae n W R 1 x y z n 1 n 1 n 1 W y ~ W z~ W R R R Moment about y-axs Moment about x-axs Moment about x-axs or y-axs ~ z 2 z ~ n ~ z 1 6 2007 S. Kenny, Ph.D., P.Eng.
Center of Mass Pont locatng the equvalent resultant mass of a system of partcles or body Generally concdes th center of gravty (G) Center of mass coordnates x m m y y ~ m m z z~ m m 7 2007 S. Kenny, Ph.D., P.Eng.
Center of Mass (cont.) Can the Center of Mass be Outsde the Body? Fulcrum / Balance Center of Mass 8 2007 S. Kenny, Ph.D., P.Eng.
Center of Gravty & Mass pplcatons Dynamcs Inertal terms Vehcle roll-over and stablty 9 2007 S. Kenny, Ph.D., P.Eng.
Geometrc Centrod Pont locatng the geometrc center of an object or body Homogeneous body Body th unform dstrbuton of densty or specfc eght Center of mass and center of gravty concdent Centrod only dependent on body dmensons and not eght terms y ~ z~ x y z 10 2007 S. Kenny, Ph.D., P.Eng.
Geometrc Centrod (cont.) Common Geometrc Shapes Sold structure or frame elements GC & CM GC & CM GC & CM Medan Lnes 11 2007 S. Kenny, Ph.D., P.Eng.
Composte Body Fnd center of gravty or geometrc centrod of complex shape based on knoledge of smpler geometrc forms 12 2007 S. Kenny, Ph.D., P.Eng.
Example 40-01 Determne the locaton (x, y) of the 7-kg partcle so that the three partcles, hch le n the x y plane, have a center of mass located at the orgn O. 1 2007 S. Kenny, Ph.D., P.Eng.
Example 40-01 (cont.) Center of Mass x y y ~ m m m m ( 7 kg) x + ( kg)( m) ( 5 kg)( 4 m) x 0 x 1.57m kg + 5 kg + 7 kg ( 7 kg) y + ( kg)( 2 m) + ( 5 kg)( 2 m) y 0 x 2.29m kg + 5 kg + 7 kg 14 2007 S. Kenny, Ph.D., P.Eng.
Example 40-02 rack s made from roll-formed sheet steel and has the cross secton shon. Determne the locaton (x,y) of the centrod of the cross secton. The dmensons are ndcated at the center thckness of each segment. 15 2007 S. Kenny, Ph.D., P.Eng.
Example 40-02 (cont.) ssume Unt Thckness Ignore bend rad Center-to-center dstance Centrod Equatons y ~ x y z 16 2007 S. Kenny, Ph.D., P.Eng. z~
Example 40-02 (cont.) Centrod Equatons x y y ~ ~ x 1 7.5mm 1 # rea (mm 2 ) ( mm) ( mm) y ~ ( mm ) y ~ ( mm ) 1 (15mm)(1mm) 15mm 2 15/2 7.5 (15)(7.5) 112.5 17 2007 S. Kenny, Ph.D., P.Eng.
Example 40-02 (cont.) Centrod Equatons x y y ~ 5 ~ y 5 25mm # rea (mm 2 ) ( mm) ( mm) y ~ ( mm ) y ~ ( mm ) 5 (50mm)(1mm) 50mm 2 50/2 25 (50)(25) 1250 18 2007 S. Kenny, Ph.D., P.Eng.
Example 40-02 (cont.) Centrod Equatons x y y ~ ~ x 6 15mm 6 ~ y 6 65mm # rea (mm 2 ) ( mm) ( mm) y ~ ( mm ) y ~ ( mm ) 6 (0mm)(1mm) 0mm 2 15 50+0/2 65 450 1950 19 2007 S. Kenny, Ph.D., P.Eng.
Example 40-02 (cont.) Centrod Equatons x y y ~ 4 6 7 5 1 2 # rea (mm 2 ) ( mm) ( mm) y ~ ( mm ) y ~ ( mm ) 1 (15mm)(1mm) 15mm 2 15/2 7.5 112.5 2 (15mm)(1mm) 15mm 2 0+15/2 7.5 562.5 (15mm)(1mm) 15mm 2 15/2 7.5 50 112.5 750 4 (0mm)(1mm) 0mm 2 15+0/2 0 80 900 2400 5 (50mm)(1mm) 50mm 2 50/2 25 1250 6 (0mm)(1mm) 0mm 2 15 50+0/2 65 450 1950 7 (80mm)(1mm) 80mm 2 45 80/2 40 600 200 Sum 25mm 2 577.5mm 9550mm 20 2007 S. Kenny, Ph.D., P.Eng.
Example 40-02 (cont.) Centrod Equatons x x 24.4mm y y ~ 577.5mm 2 25mm 9550mm 25mm 2 24.4 mm 40.6 mm y 40.6mm 21 2007 S. Kenny, Ph.D., P.Eng.
Example 40-0 To blocks of dfferent materals are assembled as shon. The denstes of the materals are: ρ 150 lb/ft and ρ 400 lb/ft. The center of gravty of ths assembly. 22 2007 S. Kenny, Ph.D., P.Eng.
Example 40-0 (cont.) Center of Gravty x y z y ~ z~ 2 2007 S. Kenny, Ph.D., P.Eng.
Example 40-0 (cont.) Center of Gravty x # y 150lb / Weght (lb) ft y ~ z z~ ( 1 / 2)( 6n)( 6n)( 2n) ( 12n / ft ) 400lb / ft ( 6n)( 6n)( 2n) ( 12n / ft ) B 16.67lb.125lb ( n) y ~ ( n) z~ ( n) ( lb n) y ~ ( lb n) z~ ( lb n).125 4 1 2 12.5.125 6.25 B 16.67 1 16.67 50 50 Σ 19.79 29.17 5.1 56.25 24 2007 S. Kenny, Ph.D., P.Eng.
Example 40-0 (cont.) Center of Gravty x 29.17 19.79 1.47n y y ~ 5.1 19.79 2.68n z z~ 56.25 19.79 2.84n 25 2007 S. Kenny, Ph.D., P.Eng.
Chapter 9 Problems Understand prncples for smple geometrc shapes Rectangle, square, trangle and crcle No closed form ntegraton knoledge requred Reve Example 9.9 and 9.10 Problems 9-44 to 9-61 Omt Example 9.1 through 9.8 Problems 9-1 through 9-4, 9-62, 9-67 to 9-8 26 2007 S. Kenny, Ph.D., P.Eng.
References Hbbeler (2007) http://ps.prenhall.com/esm_hbbeler_eng mech_1 27 2007 S. Kenny, Ph.D., P.Eng.