Transcript Vectors & Scalars

David Raju Vundi
21 July 2015
http://ap-physics.david-s.org
Scalars
 A scalar quantity is a quantity that has magnitude only
and has no direction in space
Examples of Scalar Quantities:
 Length
 Area
 Volume
 Time
 Mass
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Vectors
 A vector quantity is a quantity that has both
magnitude and a direction in space
Examples of Vector Quantities:
 Displacement
 Velocity
 Acceleration
 Force
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Vector Diagrams
 Vector diagrams are
shown using an arrow
 The length of the arrow
represents its magnitude
 The direction of the
arrow shows its direction
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Resultant of Two Vectors

The resultant is the sum or the combined effect of
two vector quantities
Vectors in the same direction:
6N
4N
=
10 N
=
10 m
6m
4m
Vectors in opposite directions:
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6 m s-1
10 m s-1
=
4 m s-1
6N
10 N
=
4N
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The Parallelogram When
Lawtwo vectors are joined

tail to tail
Complete the parallelogram

The resultant is found by
drawing the diagonal



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When two vectors are joined
head to tail
Draw the resultant vector by
completing the triangle
Problem: Resultant of 2 Vectors
Two forces are applied to a body, as shown. What is the magnitude
and direction of the resultant force acting on the body?
Solution:


Complete the parallelogram (rectangle)
The diagonal of the parallelogram ac
represents the resultant force
The magnitude of the resultant is found using
Pythagoras’ Theorem on the triangle abc
Magnitude ac  12  5
2
a
2
ac  13 N
12
Direct ionof ac : t an 
5
12
   t an1  67
5
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d
5
b

12 N
θ
5N

12
c
Resultant displacement is 13 N 67º
with the 5 N force
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Problem: Resultant of 3 Vectors
Find the magnitude (correct to two decimal places) and direction of the
resultant of the three forces shown below.
Solution:
Find the resultant of the two 5 N forces first (do right angles first)
ac  52  52  50  7.07 N
5
tan    1    45
5



Now find the resultant of the 10 N and
7.07 N forces
The 2 forces are in a straight line (45º +
135º = 180º) and in opposite directions
So, Resultant = 10 N – 7.07 N = 2.93 N
in the direction of the 10 N force
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5
d
5N

c
5
a
90º
θ
45º
135º
5N
b
Recap
 What is a scalar quantity?
 Give 2 examples
 What is a vector quantity?
 Give 2 examples
 How are vectors represented?
 What is the resultant of 2 vector quantities?
 What is the triangle law?
 What is the parallelogram law?
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Resolving a Vector Into Perpendicular
Components
 When resolving a vector into

Here a vector v is resolved into
an x component and a y
component
y
components we are doing the opposite
to finding the resultant
 We usually resolve a vector into
components that are perpendicular to
each other
x
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Practical Applications
y=25 N
 Here we see a table being
pulled by a force of 50 N at
a 30º angle to the
horizontal
 When resolved we see that
this is the same as pulling
the table up with a force of
25 N and pulling it
horizontally with a force of
43.3 N
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30º
x=43.3 N

We can see that it
would be more
efficient to pull the
table with a
horizontal force of
50 N
Calculating the Magnitude of the
Perpendicular Components
 If a vector of magnitude v and makes an angle θ with the
horizontal then the magnitude of the components are:
 x = v Cos θ
y=v Sin θ
y
 y = v Sin θ
θ

x=vx Cos θ
Proof:
x
Cos  
v
x  vCos 
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y
Sin 
v
y  vSin
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Problem: Calculating the magnitude of
perpendicular components
A force of 15 N acts on a box as shown. What is the horizontal
component of the force?
HorizontalComponent x  15Cos60  7.5 N
VerticalComponent y  15Sin60  12.99 N
12.99
Vertical
Component
N
Solution:
60º
Horizontal 7.5
Component
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N
Summary
 If a vector of magnitude v has two perpendicular
components x and y, and v makes and angle θ with the
x component then the magnitude of the components
are:
 x= v Cos θ
 y= v Sin θ
y=v Sin θ
y
θ
x=v Cosθ
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