ACS College of Engineering - MATH. 101
4. Polar coordinate system
Polar coordinate system
• Answers and detailed solutions to all problems are provided in iOS/Android "PhysOlymp" app
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...
ACS College of Engineering - MATH. 101
4. Polar coordinate system
Polar coordinate system
• Answers and detailed solutions to all problems are provided in iOS/Android "PhysOlymp" app
• With any suggestions please write to
[email protected]
Selection of fit for purpose coordinate system can significantly
simplify solution of the problem. For the cases when trajectory
of the particle is close to circular or spiral like contour in a plane,
it is usually convenient to use polar coordinates
Polar coordinate system is characterized by a distance r from the
reference point O and angle ’ from some reference direction
(Ox)
Location of some specific point A in Cartesian system in terms of
known polar coordinates can be expressed as
xA = r cos’
yA = r sin’
Components of velocity ~v in Cartesian coordinates are described
with two unit vectors ~i and ~j, which are parallel to x and y axises
respectively, with a direction alongside of increasing values of coordinates x and y
~v
x =
d x
dt
~i
~v
y =
d y
dt
~j
Total velocity in Cartesian coordinate system is a vector sum of
its components:
~v = ~v
x + ~vy = d x dt ~i + d y dt ~j
Similarly, directions in polar coordinate system are characterized
with two unit vectors: one with a normal n~ outward direction
and another one perpendicular to vector n~ with a counterclockwise direction ~τ
By definition of velocity, it is a displacement in unit of time. Let’s
consider a motion in plane from the point A to the position A0
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Normal component of velocity vn can be defined as displacement
∆r from the origin of coordinates in a small time interval ∆t
v
n =
∆r
∆t
With accounting to direction described by unit vector n~ this can
be rewritten as
~vn
=
∆t
n~ =
dt
n~ = ˙rn~ | (1)
∆r | drSimilarly tangential component of velocity can be defined as displacement r∆’ obtained by rotating at small angle ∆’ during
time interval ∆t
v
τ =
r∆’
∆t
Direction of tangential component of velocity coincides with unit
vector τ. Then,
~vτ
=
r∆’
∆t
~τ = r d’
dt
~τ = r’ ~τ ˙ (2)
Equations (1) and (2) can be easily comprehended by considering limiting cases, with trajectory of the moving particle being
either a straight line or a circle
For a motion along a straight line equation (1) becomes a regular
definition of velocity as a measure of change of coordinate r per
unit of time
v
n =
dr
dt
= ˙r
For another limiting case of the motion around a fixed center
with constant radius r, velocity of the object can be defined by
its angular velocity ! = ’˙ as
v
τ = !r = ’˙r
which corresponds to a general form of tangential component of
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