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Robertson, PhD, FCSB Biomechanics Laboratory, School of Human Kinetics, University of Ottawa, Ottawa, CanadaP}#    > /)Moment of a Force (when F is at 90 to d)X*!turning effect of a force, also called torque product of force (F) and moment arm (d) of the force from the axis (A) of rotation moment arm is the perpendicular distance from the axis of rotation to the line of the force M = F d ' j 1)Moment of a Force (when F is at 90 to d)X* !2 M = F d direction (+ / " sign) of the moment of force depends on the right-hand rule i.e., counter-clockwise is positive units are newton metres or N.mv PP      2Moment of a Force (r, F, q)0if moment arm length is difficult to compute use: M = r F sin q r is length of line from axis to line of force theta (q) is angle between line of force and line of r 2g4     3    ,   9 Moment of a Force (r, F, q)04 M = r F sin q direction (sign) of moment follows right-hand rule i.e., if force  turns line r counter-clockwise about axis at A then moment is positive   P       6Moment of a Force (r, F, q)0dto simplify and clarify positive directions of moments and forces, add reference axes to each figure<eG    ?#Moment of a Force (r, F, q)0Factors that increase moment of force increase force (F) increase lever arm length (r) increase angle (q) between lever and line of force to perpendicular M = r F sin q&PuPP&       3   5!Moment of a Force (cross-product)tif components of force and line connecting axis to line of force are known, use vector cross-product: M = r x Fgg        ;!!Moment of a Force (cross-product)6first resolve vectors r and F into their rectangular coordinates then apply: M = r x F = ( rx Fy " ry Fx ) k k is the unit vector about the Z-axisM)&    0                     $  >c+G'!Moment of a Force (cross-product)6first resolve vectors r and F into their rectangular coordinates then apply: M = r x F = ( rx Fy " ry Fx ) k k is the unit vector about the Z-axisM)&    0                     $  >c+A$!Moment of a Force (cross-product)6put components at their original points of application 76  C%!Moment of a Force (cross-product)yput components at their original points of application next slide force vectors along their lines of action and multiply  zy  ="!Moment of a Force (cross-product)if only scalar part of the moment is wanted, use this notation: M = [ r x F ]z M = ( rx Fy " ry Fx ) B+B                       >[E&Example:Given: r = (20.0, "65.0) cm F = (220, 150.0) N M = [ rx Fy " ry Fx ]z =[ (20.0 150.0) " (-65.0 220) ] = 3000 + 14 300 = 17 300 N.cm = 173.0 N.m    <P;C/(03478:<>@ B D F H PX  0` ̙33` 3` 3333f` 999MMM` f` f3` 3>?" dZ@$|? " dZ@  @" `  n?" dZ(@   @@``PR    @ ` ` p>  >   (     `Hgֳgֳ ?`8 R >*   `Xgֳgֳ ?`  X @*   `LXgֳgֳ ?`X X @*  ZXgֳgֳ ?xP X ? Slide Title  #  TLXgֳgֳ ?x X ;Body Text Second Level Third Level Fourth Level Fifth Level     <f  Nyf޽h @? ? inCa1 Biomech0 +#P(    T<`jJjJ ?P   ` R*    TA`jJjJ ?   ` T*    Z(E`jJjJ ?`P  ` R*    Z4J`jJjJ ?`  ` T*  #  TdF`gֳgֳ ? @ ` ;Body Text Second Level Third Level Fourth Level Fifth Level     <p  01 ?$  `H  0޽h? ? a(80___PPT10.S PH ( X    TpjJjJ ?P    D*Z   TljJjJ ?    F*Z   ZjJjJ ?`P   D*Z   Z8jJjJ ?`   F*Z H  0޽h? ? a(80___PPT10.e  0N0 5 0v(    BA ??n5 $ $00hhh333333ffffffffffffffffff333333  C x\`gֳgֳ ))?xP  `   C x4`gֳgֳ ))? ` H  0f޽h ? inCa1y___PPT10Y+D=' B= @B + 0N0 5   PV (     Z|`gֳgֳ ?JP  `    Z}`gֳgֳ ?N <$ 0 ` 8 {  {    `o))?"`{ ~B  ND>))?     # l `))?  4  g force (F)B Z   # l4`))? <axis (A) Z B  TDԔ))?0 C   # l`))?3 $ Kline of action of forceZB   ZDԔ))?    Z))?~  # l`))?0 S +  \moment arm (d)2Z hT z] # C  ~B  NDo))?z~B  NDo))?z]H  0f޽h ? inCa1___PPT10f+ze/D' `= @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*.D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*.D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*.D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*.D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*+8+0+0 +w  0N0 5 r j p (     ZP`gֳgֳ ?~JP  `    Zx`gֳgֳ ?] } ` ^F {   {   Zo))?"`{ xB  HD>))?     fأ`))?  4  g force (F)B Z   f\`))? <axis (A) Z ~B  NDԔ))?0 C    f`))?3 $ Kline of action of forceZB   TDԔ))?    T))?~    fP`))?0 S +  \moment arm (d)2Z hT z] # C  ~B  NDo))?z~B  NDo))?z]H  0f޽h ? inCa1y___PPT10Y+D=' = @B +{ 0N0 5 i a  (     Z`gֳgֳ ?xP  `    Z`gֳgֳ ?]~ }<$ 0 `  8 {  {    `o))?"`{ ~B  ND>))?     # l`))?  4  g force (F)B Z  # l`))? <axis (A) Z B  TDԔ))?0 C   # l``))?3 $ Kline of action of forceZB   ZDԔ))?    Z))?~    # l\`))? h  fline from A to force (r)2Z9   f``))?% P  angle between r and F (q)fZ2  # lGԔ))?  H  0f޽h ? inCa1___PPT10f++dD' `= @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*2D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*2D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2B%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*2BD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*2BD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Bq%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*BqD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*BqD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*q%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*qD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*q+8+0+0 +  0N0 5    (     Z<sgֳgֳ ?xP  s    Zsgֳgֳ ?]~ } s F {   {   Zo))?"`{ xB  HD>))?     f`s))?  4  g force (F)B Z   fs))? <axis (A) Z ~B  NDԔ))?0 C    fs))?3 $ Kline of action of forceZB   TDԔ))?    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T))?~    fSs))? h  fline from A to force (r)2Z9   fD[s))?% P  angle between r and F (q)fZ2  # lGԔ))?  dB  <DԔ))?l 4  6 o,$D 0lB  <DԔ))?MlB B <DԔ))?M M2  TԔ))?4   fbs))?v 9+Z(3  Tfsgֳgֳ ?6 Q ,$ 0 {-positive directions are defined by the arrows2.   l Q  Q ,$D 0  TLjsgֳgֳ ?Q  -add axes labels (X, Y) for additional clarity<.     @ F  F    fLTs))?FE 5YZ   fqs))?   5XZB  ZgֳgֳD>?"0@NNN?N} C,$D 0H  0f޽h ?/  inCa1___PPT10o++B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(DF' =%(D' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*+8+0+0 +8  0N0 5 {P (     Zsgֳgֳ ?xP  s    Zsgֳgֳ ?]Q <<$ 0 s F {   {   Zo))?"`{ xB  HD>))?     fs))?  4  g force (F)B Z   fs))? <axis (A) Z ~B   NDԔ))?0 C    fs))?3 $ Kline of action of forceZB   TDԔ))?    T))?~    f,s))? h  fline from A to force (r)2Z9   fs))?% P  angle between r and F (q)fZ2  # lGԔ))?  dB  <DԔ))?B  ZgֳgֳD>?"0@NNN?N` d ,$D 0B  ZgֳgֳD>?"0@NNN?N c ,$D 0B  ZgֳgֳD>?"0@NNN?N  ,$D  0H  0f޽h ? inCa1))___PPT10)+HSD/)' s= @B D(' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&9%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*&9D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*&9Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D1' =4@BB BB%(D' =-o6Bdissolve*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9W%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*9WD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*9WDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D1' =4@BB BB%(D' =-o6Bdissolve*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*W%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*WD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*WDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D1' =4@BB BB%(D' =-o6Bdissolve*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*+8+0+0 + 0N0 5  P(     `o))?"`{ |B  TDԔ))?C0    Z|sgֳgֳ ?gP  s    Z|sgֳgֳ ?]~ }<$ 0 s vB  ND>))?    # ls))?   g force (F)B Z  # ls))? o <axis (A) Z B  ZDԔ))?   Z))?~  # l@s))? M  Er(ZH  0f޽h ? inCa1  ___PPT10` ++dD ' s= @B D ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*f%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*fD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*fD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*gu%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*guD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*gu+8+0+0 +C  0N0 5 >6 (     Zsgֳgֳ ?P  s    Zhsgֳgֳ ?]$ } s    fs))? o <axis (A) Z   Zo))?"`{ vB  NDԔ))?C0 pB  HD>))?     fs))?   g force (F)B Z   fPs))? o <axis (A) Z |B  TDԔ))? |  T))?~   f s))? M  Er(ZH  0f޽h ? inCa1y___PPT10Y+D=' = @B +;  0N0 5 6 . 0 (  0 0 Zo))?"`{ vB 0 NDԔ))?C0  0  Z(gֳgֳ ?P    0  Zgֳgֳ ?]$ }  vB 0 ND>))?   0  fh ))?   g force (F)B Z 0  fhs))? o <axis (A) Z B  0 ZDԔ))? |  0 T))?~  0  f))? M  Er(Z|B  0 TD>))? g dB  0 <D>))? gg jB 0 BD>))? B 0 ZD>))?   0  f))? !  HFy*Z  0  f,))?f o bFx*Z  0  f))?    bry*Z  0 Z#))?f o brx*Z H 0 0f޽h ? inCa1y___PPT10Y+D=' = @B +?  0N0 5 E=p (    Zo))?"`{ vB  NDԔ))?C0    ZX0gֳgֳ ?P      Z01gֳgֳ ?]$ }  vB  ND>))?     f2))?   g force (F)B Z   f4))? o <axis (A) Z B   ZDԔ))? |B   TD>))? g jB  BD>))?    f>))? !  HFy*Z    f B))?f o bFx*Z    f))?S  B   `D>))?|   T))?~F8  g   g lB  <D>))?g g B  ZD>))?  H  0f޽h ? inCa1vn___PPT10N+ZD2' = @B D' = @BA?%,( < +O%,( < +D%' =%(D' =%(Du' =4@BBB B%()))D' =-6B'blinds(horizontal)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*+j  0N0 5    4 (      Zo))?"`{ vB   NDԔ))?C0     Z8^gֳgֳ ?P       Z_gֳgֳ ?]$ }  vB   ND>))?      fp`))?   g force (F)B Z    fPc))? o <axis (A) Z B   ZDԔ))?     fI))? 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""")))UUUMMMBBB999|PP3f333f3333f3ffffff3f̙3ff333f333333333f33333333f33f3ff3f3f3f3333f33̙33333f333333f3333f3ffffff3f33ff3f3f3f3fff3ffffffffff3ffff̙fff3fffff3fff333f3f3ff3ff33f̙̙3̙ff̙̙̙3f̙3f333f3333f3ffffff3f̙3f3f3f333f3333f3ffffff3f̙3f3ffffffffff!___www    """)))UUUMMMBBB999|PP3f333f3333f3ffffff3f̙3ff333f333333333f33333333f33f3ff3f3f3f3333f33̙33333f333333f3333f3ffffff3f33ff3f3f3f3fff3ffffffffff3ffff̙fff3fffff3fff333f3f3ff3ff33f̙̙3̙ff̙̙̙3f̙3f333f3333f3ffffff3f̙3f3f3f333f3333f3ffffff3f̙3f3ffffffffff!___www4#A k(kBʦ """)))UUUMMMBBB999|PP3f3333f333ff3fffff3f3f̙f3333f3333333333f3333333f3f33ff3f3f3f3333f3333333f3̙33333f333ff3ffffff3f33f3ff3f3f3ffff3fffffffff3fffffff3f̙ffff3ff333f3ff33fff33f3ff̙3f3f3333f333ff3fffff̙̙3̙f̙̙̙3f̙3f3f3333f333ff3fffff3f3f̙3ffffffffff!___www==============>=>=>=>=>=>=>=>=>>>>>>>>>>>>a>>>b>a>b>b>b>b>b>b>b>b>b?b>b?b?bbb?bbbbbbbbbbbbcbbbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcb============>==>>=>=>=>=>=>=>>>>>>>>>>>>>>a>>>b>b>b>b>b>b>b>b>b>b?b>b?b?b?b?bbb?bbb?bbb?bbb?bbcbbbcbbbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbc================>=>=>=>=>=>=>=>=>>>>>>>>a>>>a>>>b>b>b>b>b>b>b>b>b?b>b?b?bbb?bbb?bbbbbbbbcbbbcbbbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbbcbcbcbccbbcb============>>=>>=>=>=>=>=>=>>>>>>>>>>>>>>>>>>b>?>b>b>b>b>b>b>b>b?b>b?b?b?b?b?b?b?b?bbb?bbb?bbb?bbb?bbc?bbcbbbcbbbcbcbcbcbcbcbc================>=>=>=>=>=>=>=>=>>>>>>>>a>>>a>>>b>a>b>b>b>b>b>b>b?b>b?b?bbb?bbb?bbb?bbbbbbbbcbbbcbbbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcb==============>>=>>=>=>=>=>>>=>>>>>>>>>>>>>>b>>>b>b>b>b>b>b>b>b>b>b?b?b?b?b?b?b?b?bbb?bbb?bbc?bbcbbbcbbbcbbbcbcbcbcbcbcbcbcbcbc================>=>=>=>=>=>=>=>=>>>=>>>>a>>>a>>>b>a>b>b>b>b>b>b>b>b>b?b>bbb?bbb?bbbbbbbbbbbbcbbbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbbcb===========>==>>=>=>=>=>=>=>>>=>>>>>>>>>>>>>>b>>>b>b>b>b>b>b>b>b?b>b?b>b?b?b?b?b?b?bbb?bbb?bbb?bbb?bbc?bbcbbbcbbbcbcbcbcbc==============>====>>>=>=>=>=>=>=>=>>>>>>>>a>>>a>>>b>a>b>b>b>b>b>b>b>b>b?b?bbb?bbb?bbb?bbbbbbbbcbbbcbbbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcb==>>>>>>>>>>>>>>>>==>>>=>>>>=>>>>=>>>>>>=>>>>>>>>>>>>>>>>>>>>>b>>>b>>>>>>>>>>>>>>>b?b?b?b?b?bbb?bbb?bbbbbbc?bbcbbbcbbbcbcbcbcbcbcbcbc======>=>>>>>>===>>==>>>>>>>>>>>>>b>b>bbbbbbbbbbbcbbbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcb=>>>>>>>>>==>>>>>=>>=>>>>>>>>>b>>>>>>>b>>b?b>b?b?b?b?b?b?b?b?b?b?bbb?bbb?bbc?bbcbbbcbbbcbcbc======>=>>>>>>=>=>>>>>>>b>b>>>>?b>b?bbb?bbbbbbbbbbbbcbbbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcb============>>=>>=>=>=>>=>>>>>>>>>>a>>>bb>b>b>b>b>bb>b?b>b?b>b?b?b?b?bbb?bbb?bbb?bbb?bbcbbbcbbbcbcbcbcbcbc=======================>==>=>=>=>=>=>=>>>=>>>>>>>>a>>>a>a>b>b>b>b>b>b>b>b>b?b>bbb?bbb?bbbbbbbbbbbbcbbbcbbbcbcbcbcbcbcbcbcbcbcbcbcbcbcb===============>==>>=>>=>=>=>=>>>>>>>>>>>>>>a>>>b>?>b>b>b>b>b>b>b>b?b>b?b?b?b?b?b?b?b?bbb?bbb?bbb?bbc?bbcbbbcbbbc========================>=>=>=>=>=>=>=>=>>>>>>>>>>>>a>>>b>a>b>b>b>b>b>b>b>b>b?b>b?b?bbb?bbbbbbbbbbbbcbbbcbcbcbcbcbcbcbcbcbcbcbcbcb=========>===========>==>>=>=>=>=>=>=>>>>>>>>>>>>>>a>>>b>b>b>b>b>b>b>b>b>b?b>b?b?b?b?bbb?bbb?bbb?bbb?bbcbbbcbbbcbcbcbc>======>>>>>>>>>>>>>=>>>=>>>>=>>>>=>=>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>a>>>b>b>b>b>b>b>b>b>b?b>b?b?bbb?bbb?bbbbbbbbcbbbcbbbcbcbcbcbcbcbcbcbcbcbcbcb====>>>>=>>=>>>>>==>=>=>>>>>>>>>>>>>>>>>>b>?>b>b>b>b>b>b>b>b?b>b?b?b?b?b?b?b?b?bbb?bbb?bbb?bbb?bbc========>=>>>=>>>>>>=>>>>>>=>>>>>>>>>a>>>a>>>b>a>b>b>b>b>b>b>b?b>b?b?bbb?bbb?bbb?bbbbbbbbcbbbcbbbcbcbcbcbcbcbcb======>>=>>>>>==>==>>>=>=>>>=>>>>>>>>>>>>>>b>>>b>b>b>b>b>b>b>b>b>b?b?b?b?b?b?b?b?bbb?bbb?bbc?bbcbbbcbb>=>>=======================>>=>=>=>=>>=>>>=>>>>a>>>a>>>b>a>b>b>b>bmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmcbcbcbcbcbcbcbcbcb=====================>==>>=>=>=>=>=>=>>>=>>>>>>>>>>m>>>bmmmbb>b>b>bmb>b?bmb?b>bmb?b?b?m?b?b?bbb?bbb?bbb>=>==>============================>=>=>=>=>=>=>=>=>>>>>>>>a>>>a>>>b>a>bmb>bb>b>b?bmbb?bbm?bbb?mbbbbbmcbbbbcbcbcbcb=>========================>>=>>=>=>=>=>>>=>>>>>>>>>>>>m>b>m?b>b>b>b>mb>b>bmb>b?bmb??b?m?bbbbbbbbc?b>=>=>=>=>===>==========================>===>=>=>=>=>=>=>>>=>>>>>>>>a>>>a>a>b>amb>b>m>b>b?b>bm?bbb?mbbbbbmbbbbbmbbccbbcbcbcbcb=>====================>=>>==>>=>>=>=>>>=>>>>>>m>>>mb>mb>?>b>mb>b>bmb>b>bmb>?b?m?b??b?b?b?b?bbb>=>=>=>=>>=>>>=>>>=>=>>>=>>>>>>>>>>>>>>>>=>>>>>>>>>>=>=>>>>>>=>>>>=>=>=>=>>>=>>>>>>>>>>>>am>>>>b>b>b>bmb>b?m?bbb?mbbbbbmbbbbbcbbbcbcbcb=>>=>==>>=>>>>>>=>>=>>=>>=>>=>=>=>=>>>=>>>>m>>m>>b>b>b>mb>bmb>b?bmb?bmb?m?bbbb?bbb?bbb?b>>>=>=>=>=>=>>=>>>>=>>>>>=>===>=>>>=>=>=>=>=>>>=>>>>>>>>a>>ma>mb>b>b>b>b?mbb?bbm?bbbbmbbbcbbbcbbbcbcb=>=>>=>==>>==>>========>==>>=>>=>=>=>=>mm>m>m>>a>mb>?>bmbmb>b>bmb>b>m?bb?b?b?b?b?bbb>=>>>=>=>=>===>=========================>=>=>=>=>=>=>=>=>>>>>>>>am>>b>b>b>m>>mb?b>mbmbbbbbbbbcbbbcb=>=>=>>=>>=>=========================>==>>=>=>=>=>m>>>>>a>>bmb>>mb>m>mb?b?bbb?bbb?ba>>>>>>=>=>=>=>=>=>=>==>============================>=>=>=>=>=>=>=>>b>b>>m>b>mbbbbbbcbbbcbbb=>=>=>=>>=>=>======================>mmmmmmmmmmmmmmmmmmm?b>b?b?b?b?b?b>>>>>>>>>=>=>=>=>=>=>=>==============================>=>=>=>==>=m>>a>>b>>m>b>bmb>m?b?bbb?bbb?bbbbbb>>=>>>=>=>=>=>>=>=>=========================>>>=>>m>>>m>>>>>>>m>>bm>mb>bbmbb>b?b?b?b?b?b?ba>>>a>>>>>>=>>>=>=>=>=>=>=>>===========================>=>=>=>=>=>>=>=>>>m>a>>b>a>bm>bb>m>b>>mbbbb?bbbbbbbbbbbb>>>>=>=>=>=>>=>>=>=======================>==>>m>m>m>>=>>>>m>m>>>m>>b>mb>>b>b>b?b>b?b>b?b>>a>>>>>>>>>>=>=>=>=>=>=>=>==>============================>=>=>=>=>=>>>m>a>>>a>>ma>b>m>b>bm>b?b?bbb?bbb?bbb?bb>>>>>>=>>>=>=>=>=>=>=>==>========================>>=>=>m>>=>>>>>>>m>>>>>mb>>>m?>mb>b>b>b>b?b>b?b?bb>a>a>>>a>>>>>>>>>>=>=>=>=>=>=>=>===>==========================>===>=>=>=>=>==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>>>>>>>============cbcbcbcbcbcbcbcbcbcbbbcbbbcbbbbbbbbbb?bbb?bbb?b?b>>>>>>>>>>>>>>>>>>=>>==>>>>>>>======================bcbcbcbbbcbbbc?bbc?bbb?bbb?bbb?b?b?b?b>b?b>b?b>b>>b>>>>>>=>>>>>>===============cbcbcbcbcbcbcbcbcbcbcbcbcbcbcbbbcbbbbbbbbbbbbbb?b?b>b?b>b>b>b>b>b>b>b>>a>>>a>>>>>>>=>=>=>=>=>=>===>==========================bbbcbbbc?bbb?bbb?bbb?bbb?b?b?b?b?b?b?b?b>b?b>b>b>b>b>b>b>?>b>?>b>>>>>>>>>>>>>>>=>>>=>=>=>=>>=>=>===============cbcbcbcbcbcbcbcbcbcbcbcbbbcbbbcbbbbbbbbbb?bbb?bbb?b?b>b?b>b>b>b>b>b>b>b>a>b>>>a>>>>>>>>>>=>>>=>=>=>=>=>=>>==>======================bcbcbcbcbcbbbcbbbc?bbc?bbb?bbb?bbb?bbb?b?b?b?b>b?b>b>b>b>b>b>b>b>b>>>b>>>>>>>>>>>>>>>>=>=>=>=>=>=>>=>================cbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbbbcbbbbbbbbbbbbbb?bbb?b?b>b?b>b>b>b>b>b>b>b>a>b>>>a>>>>>>>>>>=>>>=>=>=>=>=>=>=>=======================bcbcbbbcbbbc?bbcbbbc?bbb?bbb?bbb?b?b?b?b?b?b>b?b>b?b>b>b>b>b>b>b>b>b>>>b>>>>>>>>>>>>>>>>=>=>=>=>=>=>>=>==============cbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbbbcbbbcbbbbbbbbbbbbbb?bbb?b?b>b>b>b>b>b>b>b>b>b>a>b>>>a>>>a>>>>>>=>>>=>=>=>=>=>=>=>===================bcbcbcbcbcbcbcbbbcbbbcbbbcbbbb?bbb?bbb?bbb?b?b?b?b?b?b>b?b>b>b>b>b>b>b>?>b>>>a>>>>>>>>>>>>>>>>=>=>=>=>>=>>=>===============bcbcbcbbcbcbcbcbcbcbcbcbcbcbcbcbcbcbbbcbbbcbbbbbbbbbb?bbb?bbb?bbb>b>b>b>b>b>b>b>b>b>a>b>>>a>>>a>>>>>>=>=>=>=>=>=>=>==>==================bcbcbcbcbbbcbbbc?bbc?bbc?bbb?bbb?bbb?b?b?b?b?b?b?b?b>b?b>b>b>b>b>b>b>b>b>>>b>>>>>>>>>>>>>>>>=>>>=>=>=>=>>=>=>===========cbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbbbcbbbbbb?bbbbbbb?bbb?b?b>b?b>b>b>b>b>b>b>b>a>b>>>a>>>a>>>>>>>>>>=>=>=>=>=>=>=>==================bcbcbcbcbcbcbcbcbcbbbcbbbcbbbcbbbb?bbb?bbb?bbb?b?b?b?b?b?b>b?b>b>b>b>b>b>b>b>b>>>b>>>a>>>>>>>>>>>>=>>>=>=>=>=>>=>=>===========cbcbbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbbbcbbbcbbbbbb?bbb?bbb?bbb>b?b>b>b>b>b>b>b>b>a>b>a>a>>>a>>>>>>=>>>=>=>=>=>=>=>>================bcbcbcbcbcbcbbbcbbbcbbbcbbbc?bbb?bbb?bbb?bbb?b?b?b?b?b?b?b?b>b?b>b>b>b>b>b>b>?>b>>>b>>>>>>>>>>>>>>>>=>=>=>=>>=>>=>============cbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbbbcbbbcbbbbbbbbbb?bbb?bbb?b?b>b?b>b>b>b>b>b>b>b>a>a>>>a>>>>>>>>>>=>=>=>=>=>=>=>==>===============bcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbbbcbbbc?bbc?bbb?bbb?bbb?b?b?b?b>b?b>b?b>b>b>b>b>b>b>b>b>>>b>>>>>>>>>>>>=>>>=>=>=>=>=>=>==>============ccbbcbccbcbcbbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbbbcbbbbbbbbbbbbbb?bbb?b?b>b?b>b>b>b>b>b>b>b>a>a>>>a>>>>>>>>>>=>=>=>=>=>=>=>===>==============՜.+,0   0 35mm Slides1' Times New RomanArial Century-WPSymbolBiomech CorelDRAW!Moments of Force*Moment of a Force (when F is at 90 to d)*Moment of a Force (when F is at 90 to d)Moment of a Force (r, F, q)Moment of a Force (r, F, q)Moment of a Force (r, F, q)Moment of a Force (r, F, q)"Moment of a Force (cross-product)"Moment of a Force (cross-product)"Moment of a Force (cross-product)"Moment of a Force (cross-product)"Moment of a Force (cross-product)"Moment of a Force (cross-product)Example  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles._9kD. Gordon E. RobertsonD. Gordon E. Robertson  !"#$%&'()*+,-/0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcefghijklmnopqrstuvwxyz{|}~Root EntrydO)PicturesZCurrent UserSummaryInformation(dPowerPoint Document(.]kDocumentSummaryInformation8