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.
/
/
/
/----------
1
a] DIAGRAM AND FILL OUT
DEFINE | ANGLE 1 |
ANGLE 2
THE FOLLOWING TABLE.
Opposite |
4 |
3
/|
SINθ
= ---------- |---------| ---
/2|
Hypotenuse | 5
| 5
/
|
----------------------------------
/
|
Adjacent |
3 |
4
5/
|
COSθ = ---------- |-----
| ---
/
|4
Hypotenuse | 5 |
5
/
|
--------------------------------
/
|
Opposite |
4 | 3
/1 |
TANθ = -------- |
--- | ---
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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