Polar Coordinates-Basic Mecanical Engineering-Lecture Slides, Slides of Mechanical Engineering

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KINEMATICS OF PARTICLES

Kinematics of particles

Rectilinear motion

Curvilinear motion

x-y coord.

n-t coord.

r-

θ

coord.

Polar coordinate system

is a two-dimensional coordinate system in which each

point on a plane is determined by a distance from a fixed point and an angle from afixed direction. The fixed point is called

the

pole

, and the ray from the pole in the fixed direction is

the

polar axis

. The distance from the pole is called the

radial coordinate

or

radius

,

and the angle is the

angular coordinate

,

polar angle

POLAR COORDINATES

3

When the particle moves in a plane (2-D), and the radial distance, r, is not constant,the polar coordinate system can be used to express the path of motion of the particle.

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In

polar

coordinates

,

a

two-dimensional

velocity

is

described by a

radial velocity

, defined as the component of

velocity away from or toward the origin

,

And angular velocity,

which is the rate of rotation about the

origin

(with

positive

quantities

representing

counter-

clockwise

rotation

and

negative

quantities

representing

clockwise rotation).

VELOCITY (POLAR COORDINATES)

5

Velocity • Instantaneous velocity

v

is obtained by the time

derivative of

r

ˆ

ˆ

=

=

=

ɺ

ɺ

r

r

r

dr

dr

v

u

ru

ru

dt

dt

• To evaluate

, note that

u

r

changes only its direction with respect to time since magnitude

of this vector = 1

r

u

^ ɺ

The instantaneous velocity is defined as:

VELOCITY (POLAR COORDINATES)

r

r

r

r

r

dr

d(ru )

ˆ

v

dt

dt

dr

du

ˆ

v

u

r

ˆ

dt

dt

v

ru

ru

ˆ

ˆ

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

ɺ

ɺ

6

r

r

ɺ

u

r

/

= u

r

u

r

u

r

= u

r

/

• u

r

u

r

∆θ

du

r

=1d

θ

d

u

r

=1d

θ

u

θθθθ

(((( )

)))

((((

))))

2

2

v

r

r

θ

=

=

=

=

ɺ

ɺ

Since

v

r

and

v

θ

are mutually perpendicular, the magnitude of

the velocity or speed is simply the positive value of

Direction of

v

is tangent to the path at P

VELOCITY (POLAR COORDINATES)

8

Direction of

v

is tangent to the path at P

Acceleration (POLAR COORDINATES)

Acceleration • Taking the time derivatives, for the instant acceleration

,

r

r

v

v u

v u

ˆ

ˆ

where

θ

θ

=

=

=

=

9

r

r

a

v

ru

ru

r

u

r

u

r

u

θ

θ

θ

θ

θ

θ

r

where v

r

and v

r

θ

θ

=

=

=

=

=

=

=

=

ɺ

ɺ

2

r

a

(r

r

)u

(r

2r

)u

θ

θ

θ

θ

Acceleration (POLAR COORDINATES)

Since

a

r

and

a

θ

are always perpendicular, the

magnitude

of the

acceleration is simply the positive value of

2

2

2

θ

θ

θ

r

r

r

r

a

Direction

is determined from the vector addition

of its components

11

of its components • Acceleration is not tangent to the path

Coordinate System •

Polar coordinate are used to solve problem involving

angular motion of the radial coordinate

r

, used to describe the

particle’s motion •

To use polar coordinates, the origin is established at a fixed

PROCEDURE FOR ANALYSIS

12

To use polar coordinates, the origin is established at a fixed

point and the radial line r is directed to the particle • The transverse coordinate

θ

is measured from a fixed

reference line to radial line

r

r

r

v

v u

v u

ˆ

ˆ

where v

r

and v

r

θ

θ θ

=

=

=

=

=

=

=

=

=

=

=

=

ɺ

ɺ

r

r

2

r

a

a u

a u

ˆ

ˆ

a

r

r

a

r

2r

θ

θ

θ

=

=

=

=

=

−

=

−

=

−

=

−

=

=

=

=

ɺ

ɺɺ

ɺɺ

ɺ

ɺ

Velocity

Acceleration

14

(((( )

)))

2

2

v

r

r

θ

=

=

=

=

ɺ

ɺ

2

2

2

r

r

r

r

a

EXAMPLE 12.

The rod

OA

is rotating in the horizontal plane such that

θ

= (

t

3

) rad

.

At the same time, the collar

B

is sliding outwards along

OA

so that

r

= (100t

2

)mm

.

If in both cases,

t

is in seconds, determine the velocity and

acceleration

of

the

collar

when

t

=

1

s

.

15

acceleration

of

the

collar

when

t

=

1

s

.

r

r

r

v

v u

v u

ˆ

ˆ

where v

r

and v

r

θ

θ θ

=

=

=

=

=

=

=

=

=

=

=

=

ɺ

ɺ

r

r

2

r

a

a u

a u

ˆ

ˆ

a

r

r

a

r

2r θ

θ

θ

=

=

=

=

=

−

=

−

=

−

=

−

=

=

=

=

ɺ

ɺɺ

ɺɺ

ɺ

ɺ

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EXAMPLE 12.

The magnitude of

v

is

(((( )

)))

((((

))))

2

2

v

r

r

θ

=

=

=

=

ɺ

ɺ

17







tan

1

2

2

−

δ δ

s

mm

v

2

2

s

mm

u

u

u r r u r r a

r

r

θ

θ

θ

θ

θ

The magnitude of

a

is

2

EXAMPLE 12.

18







tan

1

2

2

2

− φ

φ

s

mm

a

2

2

2

θ

θ

θ

r

r

r

r

a

Coordinate System.

For this unusual path, use polar coordinates.

Velocity and Acceleration.

(sin

(cos

(sin

cos

r r r

EXAMPLE 12.

20

(sin

(cos

r

Evaluating these results at

θ

2

15

. 0 0 3. 0

θ

ɺ

ɺ ɺ

ɺ

−

=

=

=

r

r

m

r

Since v = 1.2 m/s

(

)

s

rad

r

r

v

2

2

EXAMPLE 12.

21

2

2

2

2

s

rad

r

r

r

r

a