Transcript Vector Addition

Vector Addition and
Subtraction
Graphic Methods
Vectors
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Quantities having both magnitude and
direction are vector quantities.
Vectors can be represented by an arrowtipped line segment.
Examples of vectors:
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Velocity
Acceleration
Displacement
Force
Vector Terminology
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Two or more vectors acting on the same
point are said to be concurrent vectors.
The sum of 2 or more vectors is called the
resultant (R).
Any vector can be described as having both x
and y components in a coordinate system.
The process of breaking a single vector into
its x and y components is called vector
resolution.
Using the Graphic Method of
Vector Addition:
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Vectors are drawn to scale and the resultant
is determined using a ruler and protractor.
Vectors are added by drawing the tail of the
second vector at the head of the first (tip to
tail method).
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The order of addition does not matter.
The resultant is always drawn from the tail
of the first to the head of the last vector.
Any number of vectors can be added using
this method.
Example Problem
A 50 N force at 0° acts concurrently with a 20 N force at 90°.
R


R and  are equal on each diagram.
Vector Subtraction
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The process of vector subtraction is the
same as vector addition.
ur ur ur
Addition: A  B  R
ur uur ur
Subtraction: A  B R where -B has
the same magnitude as B but its
direction is reversed by 180o.
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Example Problem:
Motion in 2 Dimensions
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A boat heads east at 8.00 m/s across a
river flowing north at 5.00 m/s.
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What is the resultant velocity of the boat?
A boat heads east at 8.00 m/s across a
river flowing north at 5.00 m/s.
5.00 m/s N
8.00 m/s E
What is the resultant velocity
of the boat?
Draw to scale and
measure.
5.00 m/s N
8.00 m/s E
R = 9.43 m/s at 32°
If the river is 80.0 m wide, how long will
the boat take to cross the river?
v = d / t
t = d / v
80.0 m = 10.0 s
8.00 m/s
How far down stream will the boat
be carried in this amount of time?
v = d / t
d = vt
5.00 m/s (10.0 s)
= 50.0 m
Advantages and Disadvantages
of the Graphical Method
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Can add any
number of vectors
at once
Uses simple tools
No mathematical
equations needed
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Must be correctly
draw to scale and at
appropriate angles
Subject to human
error
Time consuming
More Vector Terminology
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Vectors are said to be in equilibrium if
their sum is equal to zero.
A single vector that can be added to
others to produce equilibrium is call the
equilibrant (E).
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E+R=0
Equal to the resultant in magnitude but
opposite in direction.
E=-R
E=5N
at 180 °
R=5N
at 0°
Solving Vectors Using the
Analytical Method
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A rough sketch of the vectors is drawn.
The resultant is determined using:
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Algebra
Trigonometry
Geometry
Quick Review
Right Triangle
c is the hypotenuse
c2 = a2 + b2
B
c
sin = o/h
a
C
A
b
cos = a/h
A + B + C = 180°
B = 180° – (A + 90°)
tan A = a/b
tan B = b/a
tan = o/a
These Laws Work for
Any Triangle.
A + B + C = 180°
C
Law of sines:
a
b
a
=
sin A
B
b
sin B
=
c
sin C
A
c
Law of cosines:
c2 = a2 + b2 –2abCos C
Use the Law of:
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Sines when you
know:
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2 angles and an
opposite side
2 sides and an
opposite angle
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Cosines when you
know:
2 sides and the angle
between them
For right triangles:
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Draw a tip to tail sketch first.
To determine the magnitude of the
resultant
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Use the Pythagorean theorem.
To determine the direction
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Use the tangent function.
To add more than two vectors:
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Find the resultant for the first two
vectors.
Add the resultant to vector 3 and find
the new resultant.
Repeat as necessary.
Advantages and Disadvantages
of the Analytical Method
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Does not require
drawing to scale.
More precise
answers are
calculated.
Works for any type
of triangle if
appropriate laws are
used.
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Can only add 2
vectors at a time.
Must know many
mathematical
formulas.
Can be quite time
consuming.
Solving Vector Problems
using the Component Method
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Each vector is replaced by 2 perpendicular
vectors called components.
Add the x-components and the y-components
to find the x- and y-components of the
resultant.
Use the Pythagorean theorem and the
tangent function to find the magnitude and
direction of the resultant.
Vector Resolution
y = h sin 
h

y
x = h cos 
x
-+
++
--
+-
Components of Force:
y
x
Example:
x
6 N at 135°
5 N at 30°
y
5 cos 30° = +4.33
5 sin 30° = +2.5
6 cos 45 ° = - 4.24
6 sin 45 ° = + 4.24
+ 0.09
R=
(0.09)2 + (6.74)2
 = arctan 6.74/0.09
= 6.74 N
= 89.2°
+ 6.74
Advantages of the
Component Method:
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Can be used for any
number of vectors.
All vectors are added at
one time.
Only a limited number of
mathematical equations
must be used.
Least time consuming
method for multiple
vectors.
Solve the following problem using
the component method.
6 km at 120
10 km at 30