Engineering Mechanics: Statics in SI Units, 12e
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1 Engineering Mechanics: Statics in SI Units, 1e 9 Center of Gravity and Centroid 1 Chapter(Objectives Concept)of)the)center)of)gravity,)center)of)mass,)and)the) centroid))) Determine)the)location)of)the)center)of)gravity)and)centroid) for)a)system)of)discrete)particles)and)a)bo)of)arbitrary) shape Chapter(Outline 1. Center)of)Gravity)and)Center)of)Mass)for)a)System)of)Particles). Composite)Bodies
2 9.1(Center(of(Gravity(and(Center(of(Mass(for(a( Center)of)Gravity)) An)object)is)composed)of)infinitesimal)bodies)and)let)each)bo)has)the) weight)dw)) Let)the)bo)is)in)the)3D)in)the)x,)y,)z)frame) The)total)weight)of)all)the)bodies)can)be)represented)by)a)single)force)W) at)point)g) 3 9.1(Center(of(Gravity(and(Center(of(Mass(for(a( Center)of)Gravity)(use)equivalent)force)and)moment)) Resultant)weight)of)an)object))the)sum)of)all)infinitestimal)weights)ofall) bodies) F R F W dw Resultant)moment)about)any)axis)The)sum)of)moments)of)all)bodies) about)the)axis) M R M Let)G)is)the)location)of)the)equivalent)force,)which)is)equal)to x W ~ x W ~ x W... ~ x W 1 1 n n y W ~ y W ~ y W... ~ y W 1 1 n n z W ~ z W ~ z W... ~ z W 1 1 n n 4
3 9.1(Center(of(Gravity(and(Center(of(Mass(for(a( Center)of)Gravity)) In)summary,)for)discrete)quantity) ~ xw i i ~ yiwi ~ ziw x ; y, z W W W i i i i For)continuous)quantity x ~ xdw ~ ydw ; y, z dw dw ~ zdw dw 5 9.1(Center(of(Gravity(and(Center(of(Mass(for(a( Center)Mass))) Using)the)relationship)W))mg,)consider)the) centre)of)mass)with)contant)g) In)general,)the)centre)of)mass)is)the)centre)of) gravity) The)centre)of)mass)is)usually)used)in)) Dynamics)problems 6
4 9.1(Center(of(Gravity(and(Center(of(Mass(for(a( Centroid)of)a)volume)) For)an)object)with)uniformly) distributed)mass,)hence) ))))))))))))))))))))or))))))))))))))))))) m ρv dm ρdv Hence)the)centre)of)volume)is 7 9.1(Center(of(Gravity(and(Center(of(Mass(for(a( Centroid)of)an)Area)) For)an)are)with)the)same)thickness)thoughout)the)area) The)area)can)be)subdivide)into)dA 8
5 9 9.1(Center(of(Gravity(and(Center(of(Mass(for(a( Centroid)of)a)Line)) For)an)object)with)a)line)shape,)consider)a)small)length)dL) From)Pythagorus L L L L dl ydl y dl xdl x ~ ; ~ ( ) ( ) dl 1 dl 1 or 10 Example(9.1 Locate)the)centroid)of)the)rod)bent)into)the)shape)of)a) parabolic)arc.
6 Example(9.1 Differential)element) dl)is)located)on)the)curve)at)the)arbitrary)point)(x,)y)) Area)and)Moment)Arms) For)differential)length)of)the)element)dL) Since)x)y )and)then)/y The)centroid)of)a)small)element)is)located)at 11 Example(9.1 Integrations 1
7 Example(9. Locate)the)centroid)of)the)circular)wire)segment)shown. 13 Example(9. Polar)coordinate)is)more)suitable)since)the)arc)is)circular) Differential)element) A)circular)arc)is)selected,)it)intersects)the)curve)at Area)and)Moment)Arms) The)differential)length)of)the)element) The)centroid)of)a)small)element)is)located)at) ~ x R cosθ and ~ y R sinθ Integration dl Rdθ ( R,θ )
8 Example(9.3 Determine)the)distance))))measured)from)the)x)axis)to)the) y centroid)of)the)area)of)the)triangle)shown. 15 Example(9.3 Differential)element) A)rectangular)element)with)thickness),)intersect)at)(x,y)) b Area)and)Moment)Arms) da x ( h y) h The)area)of)the)element)) ~ y y The)centroid)of)a)small)element)is)located)at) Integration y ~ yda 0 y ( b h)( h y) ( b h)( h y) ( 1 6) bh ( 1 ) bh 3 A 0 h h A da h 16
9 Example(9.4 Locate)the)centroid)for)the)area)of)a)quarter)circle)shown. 17 Example(9.4 Polar)coordinate)is)used.) Differential)element) A)triangle)element)with)angle))))))))intersect)at)) θ Area)and)Moment)Arms) The)area)of)the)element)) The)centroid)of)the)triangle)element)is)located)at)(Ex.)9.3)) Integration da d ( R,θ ) ( 1 )( R)( Rdθ ) ~ x ~ R ( 3) Rcosθ and y ( 3) sinθ
10 Example(9.5 Locate)the)centroid)for)the)area)shown)below. 19 Example(9.5 Solution)I)(Fig.)a)) Differential)element) A)rectangular)element)with)thickness),)has)a)height)y.) Area)and)Moment)Arms) The)area)of)the)element)) The)centroid)is)located)at) Integration da y ~ x x, and ~ y y 0
11 Example(9.5 Solution)II)(Fig.)b)) Differential)element) A)rectangular)element)with)thickness),)has)a)length)(1Wx).) Area)and)Moment)Arms) The)area)of)the)element)) da ( 1 x) The)centroid)is)located)at) ~ x (1 x) / (1 x) / Integration x, and ~ y y 1 9.(Composite(Bodies( is)composed)of)objects)with)simple)geometries) such)as)triangles,)rectangles,)circles) Consider)individual)geometries)to)find)the)centroid x ~ xw W y ~ yw W z ~ zw W
12 9.(Composite(Bodies Analysis)method) Composite)Parts) Devide)a)big)object)in)to)a)series)of)simple)geometries) For)the)void)or)spaces,)the)geometries)have)negative)values) Moment)Arms) Locate)the)frame)and)then)find)the)centroid)for)each)simple) geometry) Summations) Find)the)centroid)the)previous)centroid)equaiton 3 Example(9.10 Locate)the)centroid)of)the)plate)area. 4
13 Solution Composite)Parts) Plate)divided)into)3)segments.) Area)of)small)rectangle)considered) negative. 5 Solution Moment)Arm) Location)of)the)centroid)for)each)piece)is)determined)and) indicated)in)the)diagram.) Summations 6
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