Coplanar Vectors and Cartesian Representa#on

Coplanar  Vectors  and   Cartesian  Representa1on  

Op#mist  –  The  glass  is  half  full.   Pessimist  –  The  glass  if  half  empty.   Engineer  –  The  glass  is  overdesigned.  

Review   ¢  Two  forces  can  be  combined  using  the  

parallelogram  law  to  form  a  resultant   ¢  A  resultant  can  be  broken  up  into  its   components  using  the  geometry  of  the  system   and  some  trig  

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Cartesian Coordinates

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Forces  and  vector  components   ¢  Two  forces  act  on  

the  hook.  Determine   the  magnitude  of  the   resultant  force.  

To solve the problem we utilized the parallelogram law. We moved the 500 N force, maintaining its orientation, until the tail of the 500 N force was positioned on the head of the 200 N force. We then solved for the angle between the two vectors at this point of contact and using the law of cosines solved for the resultant vector.

Cartesian Coordinates

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Forces  and  vector  components   ¢  What  would  the  magnitude  of  the  600N  

force  have  to  change  to  so  that  the   resultant  of  the  two  forces  would  align   along  the  posi1ve  x-­‐axis?  

We considered the components of each of the forces along the positive x-axis (the i component) and the positive y-axis (the j-component). The problem condition states that the resultant cannot have a j-component so the sum of the jcomponents of the two vectors which are being added together must be equal to 0. Using a sign convention that a positive direction is toward the labeled end of the axis, we resolve the unknown force (represented by the 600 N vector in the drawing) into its x and y components and set the magnitude of the y-component equal to 800 N. This allows us to solve for the magnitude of the unknown vector. Cartesian Coordinates

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Cartesian  Coordinates   ¢  There  is  a  special  case  when  two  components  

that  are  perpendicular  are  combined  

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Cartesian Coordinates

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Cartesian  Coordinates   ¢  The  resultant  of  these  two  vectors,  F,  is  

formed  in  the  usual  manner  by  connec1ng  the   tail  of  the  sta1onary  vector  to  the  head  of  the   moved  vector   y

   F = F1 + F 2

F

F1 x

F2 6

Cartesian Coordinates

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Cartesian  Coordinates   ¢  U1lizing  Cartesian  coordinates.  

   F = F1 + F 2

y

F

Fy

F = F +F 2 x

2 y

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x

Fx

Cartesian Coordinates

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Cartesian  Coordinates   ¢  From  the  drawing,  we  have  the  following  

rela1onships  

tan (α ) =

Fy Fx

y

Fy = F sin (α )

Fx = F cos (α ) 8

Cartesian Coordinates

F α Fx

Fy x 31 August 2012

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Resultant  of  a  Series  of  Vectors   ¢  The  really  nice  part  of  this  comes  when  we  

take  a  series  of  forces  at  a  point  and  develop  a   single  resultant  from  all  the  forces  

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Cartesian Coordinates

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Resultant  of  a  Series  of  Vectors   ¢  We  will  start  with  three  forces  F1,  F2,  and  F3  

and  try  and  find  the  resultant,  F  which  is  the   vector  sum  of  the  three  forces   y

F3

F2 x

F1 10

Cartesian Coordinates

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Resultant  of  a  Series  of  Vectors   ¢  We  have  already  moved  (or  the  forces  were  

already  posi1oned)  so  that  their  tails  were  at   the  same  point   y

F3

F2 x

F1 11

Cartesian Coordinates

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Resultant  of  a  Series  of  Vectors   ¢  The  resultant  F  is  the  vector  sum  of  the  three  

forces,  F1,  F2,  and  F3  

    F = F1 + F 2 + F 3

y

F3

F2 x

F1 12

Cartesian Coordinates

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Resultant  of  a  Series  of  Vectors   ¢  Considering  the  two  forces  in  components.  

( ) = F sin ( β ) = F cos (α ) = F sin (α )

y

F1x = F1 cos β F1y

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F2x F2y 13

F3

F2

F1x

2

F1y

2

Cartesian Coordinates

F2y

α

F2x

β

F1

x

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Resultant  of  a  Series  of  Vectors   ¢  The  components  of  F3  

y

F3 F3y

γ

x F3x

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Cartesian Coordinates

F4 31 August 2012

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Resultant  of  a  Series  of  Vectors   ¢  And  subs1tu1ng  our  trigonometric  evalua1ons  

Fx = F2 cos (α ) − F1 cos ( β ) − F3 cos (γ ) Fy = F2 sin (α ) − F1 sin ( β ) + F3 sin (γ ) y

F3 F3y

γ

x 15

F3x Cartesian Coordinates

F4

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Resultant  of  a  Series  of  Vectors   ¢  If  we  maintain  a  consistent  sign  conven1on,  

we  can  make  a  general  statement  about   finding  the  resultant  of  any  n  vectors  whose   lines  of  ac1on  intersect  at  a  point  

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Cartesian Coordinates

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Resultant  of  a  Series  of  Vectors   n

Fx = ∑ Fxi i =1 n

Fy = ∑ Fy i i =1

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Cartesian Coordinates

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Resultant  of  a  Series  of  Vectors   ¢  Remember  that  this  is  only  true  if  we  maintain  

a  consistent  sign  conven1on  and  take  the   algebraic  sign  from  our  axis  selec1on   n

Fx = ∑ Fxi i =1 n

Fy = ∑ Fy i i =1

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Problem   Determine the magnitude of the resultant force acting on the corbel and its direction θ measured counterclockwise from the x-axis.

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Cartesian Coordinates

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Problem   If the magnitude of the resultant force acting on the eyebolt is 600 N and its direction measured counterclockwise from the positive x-axis is θ = 30°, determine the magnitude of F1 and the angle ϕ.

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Homework   ¢  Remember  that  you  need  to  have  the  

“Clicker”  with  you  on  Wednesday   ¢  Problem  2-­‐35   ¢  Problem  2-­‐41   ¢  Problem  2-­‐50  

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