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Ch 6 Vectors

Updated 5/26/03
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 VECTOR ADDITION RULES

PARALLELOGRAM RULE:  Draw the vectors tail to tail, construct the parallelogram, the sum of two vectors is the diagonal.  

VECTOR MOVEMENT RULES:
a] Vectors can be moves if you don't change their length or angle.
b] ANY NUMBER of vectors can be added by drawing them head to tail.  
c] The resultant is drawn from the tail of the 1st vector to the head of the last.

7  a] VECTORS CAN BE MOVED IF YOU DON'T CHANGE THEIR SIZE OR DIRECTION & PLACE THEM 
      HEAD TO TAIL.
   b] VECTORS CAN BE ADDED BY DRAWING THE FINAL VECTOR FROM THE TAIL OF THE 1st 
      VECTOR TO HEAD OF THE LAST VECTOR.
   c] TWO VECTORS CAN BE SUBTRACTED BY REVERSING THE DIRECTION OF THE SECOND VECTOR 
      AND THEN ADD THEM:  ΔV = V1 + (-V2)
   d] THE SUM OF ANY 2 VECTORS IS THE DIAGONAL OF THE PARALLELOGRAM.
   e] AS THE ANGLE BETWEEN 2 VECTORS INCREASES THEIR SUM DECREASES.
   f] THE SUM OF ANY SET OF VECTORS IS CALLED THE RESULTANT.
   g] A VECTOR CAN BE RESOLVED INTO ITS COMPONENTS.
   h] 2 OR MORE FORCES ACTING AT THE SAME TIME ON THE SAME POINT ARE CONCURRENT.
   i] RESOLUTION OF FORCES IS THE PROCEDURE FOR FINDING THE COMPONENTS OF A FORCE.  

Angle check:

If V1 = V2, Then  θ =  Original Angel /2   See diagram on board.
If V1 > V2, Then  θ < Original Angle /2    See diagram on board.
If V2 > V1, Then  θ > Original Angle /2    See diagram on board.

      /  
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/----------

 

1    a] DIAGRAM AND FILL OUT           DEFINE   | ANGLE 1 | ANGLE 2
THE FOLLOWING TABLE.            Opposite   |    4    |  3
        /|               SIN
θ = ---------- |---------| ---
       /2|                      Hypotenuse |    5    |  5
      /  |               ----------------------------------
     /   |                      Adjacent   |  3      |  4
   5/    |               COS
θ = ---------- |-----    | ---
   /     |4                     Hypotenuse |  5      |  5
  /      |               --------------------------------
 /       |                      Opposite   |      4  |  3
/1       |               TAN
θ = --------   |     --- | ---
    3                           Adjacent   |      3  |  4
b] WHAT DOES EACH SYMBOL MEAN? 
θ = SIN-1 Rv/R
   SIN
-1 Rv/R  is the angle whose SIN = Rv/R
   COS
-1 Rh/R  is the angle whose COS = Rh/R
   TAN
-1 Rv/Rh is the angle whose TAN = Rv/R
   Rv = Vertical component of the Resultant
   Rh = Horizontal component of the Resultant
   R  = the Resultant

 2   AT 18 m/s A BOAT HEADS ACROSS A 200 m WIDE RIVER FLOWING AT 15 m/s.  FIND:  a] THE MAGNITUDE & DIRECTION OF THE BOAT'S RESULTANT VELOCITY.
       b] HOW LONG IT TAKES TO REACH THE OTHER SIDE.
       c] HOW FAR DOWN STREAM IT IS WHEN IT REACHES THE OTHER SIDE.
P:  Boat = 20 m/s, 300 m, 6 m/s   HMWK:  Boat = 22 m/s, 180 m, 12 m/s

 3 SKETCH & MATHEMATICALLY RESOLVE A 140 lb FORCE INTO ITS COMPONENTS IF IT IS ACTING AT 30o WITH THE HORIZONTAL. 
a] SKETCH & LABEL THE PROBLEM    b] SHOW EQ & SUBSTITUTION, & ANSWERS.
c] SHOW SIZE CHECK & DRAW ANGLE CHECK DIAGRAM. PRACT 45o.  HMWK:  60o

 4 SKETCH (Show Rv,Rh,R) & MATHEMATICALLY ADD THE FOLLOWING VECTORS:
a] A 60 lb & A 80 lb FORCE ACTING AT 45o.
b] SHOW THE SIZE & ANGLE CHECKS.
PRACT: A PLANE THAT TRAVELS 120 mi/h EAST IN A 90 mi/h 30o NE WIND.
HMWK:  AIR PLANE AT 150 mi/h EAST IN A 60 mi/h 45o NE WIND.

 5 SKETCH BEFORE & AFTER DIAGRAMS (Show Rv,Rh,R) & FIND THE RESULTANT.
  VECTOR | MAGNITUDE | DIRECTION       VECTOR  | MAGNITUDE | DIRECTION
---------|-----------|----------      ---------|-----------|----------
a]  F1   |    50 lb  |    20o        b]   F1   |   60 lb   |     0o  
    F2   |    50 lb  |    40o             F2   |   70 lb   |    40o  
    F3   |    50 lb  |    60o             F3   |   55 lb   |    70o  
---------|-----------|----------      ---------|-----------|----------
---------|-----------|----------      ---------|-----------|----------
c]  F1   |   75 lb   |    30o         HMWK F1  |    60 lb  |    15o  
    F2   |   45 lb   |    60o              F2  |    60 lb  |    45o 
    F3   |   60 lb   |   135o              F3  |    60 lb  |    75o 
         |           |                     F4  |    60 lb  |   105o 
---------|-----------|----------      ---------|-----------|----------

 6   A WOMAN TIES A 20 ft STEEL CABLE FROM HER CAR TO A TREE, PUSHES THE MIDDLE WITH 100 lb AND DEFLECTS THE CABLE 1 ft. 
DIAGRAM AND FIND THE FORCE EXERTED ON THE CAR. 
PRACT:  15 ft, 100 lb, 0.5 ft HMWK:  25 ft, 80 lb, 2 ft

 7   400 N TRUNK IS PLACED ON AN INCLINED PLANE AT A 40o ANGLE.
a] DRAW THE VECTOR DIAGRAM & FIND THE:    d] NET FORCE;
b] NORMAL F| & PARALLEL COMPONENTS F||    e] TRUNK'S MASS.
c] FRICTION FORCE,
m = 0.50   f] ACCELERATION OF THE TRUNK
 P1: 350 N, 20o, 0.30  P2: 375 N, 25o, 0.55  HMWK: 400 N, 10o, 0.25

 8   a] Diagram the trunk on the incline plane at  0o angle.
b] Diagram the trunk on the incline plane at 90o angle.
c] As the inclined plane becomes steeper the F| _?_ and F|| _?_.
d] DEFINE, & DERIVE THE EQUATION FOR STATIC FRICTION,
m.

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