Physics 231
General University Physics
Spring 2007
Lecture 5
Last Lecture
Constant acceleration
Kinematic equations
Today
Properties of Vectors
Components
Adding Vectors Algebraically
Coordinate Systems
Used to describe the position of a point in space
Coordinate system consists of
a fixed reference point called the
origin
specific axes with
scales
and
labels
instructions on how to label a point relative to
the origin and the axes
Cartesian Coordinate System
Also called rectangular
coordinate system
x
 and
y
 axes intersect
at the origin
Points are labeled (
x
,
y
)
Scales
(m, cm, mi, etc.)
Rene Descartes
(cm)
(cm)
Polar Coordinate System
Origin and reference
line are noted
Point is distance
r
from
the origin in the
direction of angle
θ
,
ccw from reference
line
Points are labeled (
r
,
θ
)
Scales
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Polar to Cartesian Coordinates
Based on forming
a right triangle
from
r
and
θ
x
=
r
cos
θ
y
=
r
sin
θ
Cartesian to Polar Coordinates
r
is the hypotenuse and
θ
an angle
θ
must be counterclock
wise (ccw) from positive
x
axis for these equations to
be valid
2
2
tan
y
x
r
x
y
θ
=
=
+
Example 3.1
The Cartesian coordinates of a
point in the
xy
plane are (
x,y
) =
(3.50, 2.50) m, as shown in
the figure. Find the polar
coordinates of this point.
Solution:
From Equation 3.4,
and from Equation 3.3,
2
2
2
2
(
3.50 m)
(
2.50 m)
4.30 m
r
x
y
=
+
=
−
+ −
=
2.50 m
tan
0.714
3.50 m
216
y
x
θ
θ
−
=
=
=
−
=
°
Note:
not 36
D
Vectors and Scalars
A
scalar quantity
is
completely specified by a
single value with an
appropriate unit and has no
direction.
A
vector quantity
is
completely described by a
number and appropriate
units plus a direction.
Vector Notation
When handwritten, use an arrow:
When printed, will be in bold print:
A
When dealing with just the magnitude of a vector in print,
an italic letter will be used:
A
or

A

The magnitude of the vector has physical units
The magnitude of a vector is always a
positive
number
A
G
Vector Example
A particle travels from A
to B along the path shown
by the dotted red line
This is the
distance
traveled
and is a scalar
The
displacement
is the
solid line from A to B
The displacement is
independent of the path
taken between the two
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 Spring '08
 ELLIS
 Physics, Cartesian Coordinate System, Acceleration, Ay, Polar coordinate system, Vector Notation

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