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VECTORS
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Scalars are quantities which are fully
described by a
magnitude alone.
Vectors are quantities which are fullydescribed by both a
magni tude and
a direction.
Do you remember the difference
between a scalar and a vector?
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1. The length of the line represents the
magnitude and the arrow indicates the
direction.2. The magnitude and direction of the vector is
clearly labeled.
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The magnitude of a vector in a scaled vectordiagram is depicted by the length of the arrow.
The arrow is drawn precisely to length in
accordance with a chosen scale.
Scaling!!!
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Direction!!!
Sometimes vectors will be directed
due East or due North. However we
will encounter vectors in all sorts ofdirections and be forced to find the
angle!
Compass Coordinate System
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Compass Coordinate System
x = 30 m @ 20 E of N
V = 20 m/s @ 30 W of N
a = 10 m/s2@ 40 W of S
F = 50 N @ 10 S of E
Navigational System?
Use the same scale for all vector magnitudes
E
S
N
W
To Draw direction:
Ex. 20 E of N: Start w/ North and go 20 East
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All these planes have the same reading on
their speedometer. (plane speed not speed
with respect to the ground (actual speed)
What
factor is
affecting
theirvelocity?
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A B C
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Easy Adding
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The resultant is the
vector sum of two or
more vectors.
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1. Select an appropriate scale (e.g., 1 cm = 5 km)
2. Draw and label 1st vector to scale.
*The tail of each consecutive vector begins at the head of
the most recent vector*
3. Draw and label 2nd vector to scale starting at the head of
the 1st vector.
4. Draw the resultant vector (the summative result of the
addition of the given vectors) by connecting the tail of the1st vector to the head of the 2nd vector. (initial to final pt.)
5. Determine the magnitude and direction of the resultant
vector by using a protractor, ruler, and the indicated
scale; then label the resultant vector.
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A. Tailwind(with the wind)
120 km/h20 km/h
80 km/h
100 km/h
100 km/h
20km/h
=
B. Headwind(against the wind)
=
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80 km/h
100 km/h
60 km/h
Resultant
C. 90 crosswindUsing a ruler and your
scale, you can determine
the magnitude of the
resultant vector. Or youcould use the
Pythagorean Theorem.
Then using a protractor,
you can measure the
direction of the resultant
vector. Or you could use
trigonometry to solve for
the angle.
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1. Find the resultant force vector of the two forcesbelow. 25 N due East, 45 N due South
25 N, East
45 N, South
Decide on
a scale!!!
51 N
59 S of E51 N
31 E of S
A i l i fl i 200 h t 50o
N f E Wi d l it
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0o
N
180o
270o
An airplane is flying 200mph at 50o
N of E. Wind velocity
is 50 mph due S. What is the velocity of the plane?
Scale: 50 mph = 1 inch
A i l i fl i 200 h t 50o
N f E Wi d l it
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E
N
W
S
An airplane is flying 200mph at 50o
N of E. Wind velocity
is 50 mph due S. What is the velocity of the plane?
Scale: 50 mph = 1 inch
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E
N
W
S
An airplane is flying 200mph at 50o
N of E. Wind velocity
is 50 mph due S. What is the velocity of the plane?
Scale: 50 mph = 1 inch
200 mph
50 mph
VR= 165 mph
@ 40 N of E
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2. Find the resultant velocity vector of the twovelocity vectors below.700 m/s @35 degrees E of N; 1000 m/s @ 30 degreesN of W
V1Vr
V2
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A bear walks one milesouth, then one milewest, and finally walksone mile north. Afterhis brisk walk, the bearends back where he
started.What color is thebear???
Intro to Vectors Warm-up
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In what direction is the leash pulling
on the dog?
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Wh t ld h t th d
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What would happen to the upward
and rightward Forces if the Force on
the chain were smaller?
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1) Find the resultant
Magnitude:__________
of the two vectors Direction:___________
Vector #1 = 20.5 N West
Vector #2 = 14.3 N North
Vector Diagram
24.99 N
34.90 N of W
V2
V1
VR=?
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2) Find the component of the
resultant = 255m 27 South of East
Vector # 1 _______ Direction__________
Vector # 2________ Direction__________
Vector Diagram:
115.8 m
227.2 m
South (-)
East (+)
Vr
V2
V1
Conventions:
+
+
-
-
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1. Draw vector diagram. (Draw axis)
2. Resolve vectors into components using trig:
Vadj = V cos Vopp = V sin 3. Sum x and y components:
Vxi Vyi4. Redraw!! Determine resultant vector using
Pythagoreans Theorem and trig:
Magnitude= ( Vxi) + ( Vyi)
Direction Action: = tan-1(opp/adj)
Practice: Find FR=Fnet =?
200 N due South,
100 N at 40 N of W
Answer: Fnet = N
@ W of S
Skip
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An airplane flies at an engine speed of 100 m/s
at 50 W of S into a wind of 30 m/s at 200 E ofN. What is the airplanes resultant velocity?
Solve using the components method!!
How far has the plane traveled after 1 hr??
Answer: 75.52 m/s @ 28.54 S of W
168.89 miles per 1 hour
You Try! ! !
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You Try! ! !
A motor boat traveling 4.0 m/s, East
encounters a current traveling 3.0 m/s,
North.
a. What is the resultant velocity of the motor boat?
b. If the width of the river is 80 meters wide, then how
much time does it take the boat to travel shore to
shore?
c. What distance downstream does the boat reach the
opposite shore?
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Equilibrium
Balanced? or Unbalanced?
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Balanced? or Unbalanced?
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Forces are in Equilibrium
A th Obj t i E ilib i ?
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Are these Objects are in Equilibrium?
50 N
151 N
25 N
15 N
10 N
102 N
5 kgCeiling
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The First Condition of Equilibrium: The upward forces must
balance the downward forces, and the leftward forcesmust balance the rightward forces.
SFx = 0 SFy = 0 at equilibrium
Solution of Problems in Static's:
1) Isolate a body. What point or object are you going to talk about?
2) Draw the forces acting on the body you have isolated, and label
them. (If their value is not know, give them a symbol
such as F1, FP, T1, etc.) Rememberthis is a free-body diag.
http://eta.physics.uoguelph.ca/cgi-bin/imagemap/ship7/27/2019 Vector Final 2007
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3) Split each of the forces into its x and y
components, and label the components in termsof the symbols given in rule 2 and the proper sines
and cosines.
4) Write down your summation equations for the first
condition of equilibrium.
5) Solve the equations for the unknowns.
Cont.
Equilibrium problem
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Equilibrium problem
10 kg
300 N
Fy
Solve for Fx & Fy?
Fx
20
Ans: Fx = 102.61 N Fy = 183.91N
Equilibrium problem
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Equilibrium problem
Solve for F?
10 kg
200 N
50 NF
30
Answer: F = 88.34 N @ 60.65 S of W
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Tension Warm-up
Find the tension in eachrope if the weight (W) is
50 N.
Be sure to pick an
appropriate point to draw
your free-body-diagram.
Then sum your x forces
and then your y-forces.
W
37 45
Answers: T1= N, T2= N
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When the strongman suspends the10N telephone book with the ropeheld vertically the tension in eachstrand of rope is 5 N. If thestrongman could suspend the bookfrom the strands pulled horizontallyas shown, the tension in eachstrand would be:
a) About 5 Nb)About 10 Nc) About 20 Nd)More than a million Newtons...basically impossible.
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The two cars below are both accelerated by a 10 N force. Whichcar experiences the larger acceleration?a) Car 1b) Car 2c) Both cars accelerate similarly.
Explain
10
10
VROOM!
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