Curvilinear Motion and Polar Coordinates

6
KINEMATICS OF PARTICLES Kinematics of particles is the study about the motion of particles - specifically to determine their d i s p I a c e m e nt, v e I o c ity and a c c e I e r at i o n. o Displacement: position relative to a reference point or coordinate. o velocity: rate of change of displacement with respect to time. o Acceleration: rate of change of velocity with respect to time. Motion : The motion of a parlicle and c urv il ine ar mot i o n. o Rectilinear motion: c Curvilinear motion: may take place in two main forms, namely rectilinear motion The particle moves in a straight line. The particle moves at random - i.e. a combination of linear and rotation. Rectilinear Motion In this motion, displacement, velocity and depending on the direction of motion Motion inX-direction : Motion in l-direction : Displacement : Velocity: Acceleration : acceleration are indicated in terms x or !, Displacement: Velocity: Acceleration : x v*= i ar= t v v, = j' an= !

description

dynamic

Transcript of Curvilinear Motion and Polar Coordinates

Page 1: Curvilinear Motion and Polar Coordinates

KINEMATICS OF PARTICLES

Kinematics of particles is the study about the motion of particles - specifically todetermine their d i s p I a c e m e nt, v e I o c ity and a c c e I e r at i o n.

o Displacement: position relative to a reference point or coordinate.

o velocity: rate of change of displacement with respect to time.

o Acceleration: rate of change of velocity with respect to time.

Motion :

The motion of a parlicleand c urv il ine ar mot i o n.

o Rectilinear motion:

c Curvilinear motion:

may take place in two main forms, namely rectilinear motion

The particle moves in a straight line.

The particle moves at random - i.e. a combination of linearand rotation.

Rectilinear Motion

In this motion, displacement, velocity anddepending on the direction of motion

Motion inX-direction :

Motion in l-direction :

Displacement :

Velocity:

Acceleration :

acceleration are indicated in terms x or !,

Displacement:

Velocity:

Acceleration :

x

v*= iar= t

vv, = j'

an= !

Page 2: Curvilinear Motion and Polar Coordinates

Curvilinear Motion

In this motion, displacement, velocity and acceleration are measured by means ofCartesian coordinates (x-y) or Polar coordinates (r-0).

Cartesian coordinates

The displacement of the particle isresolved into two components, i.e. onealong the OX axis and the other alongthe OIaxis, as shown on the right.

The velocity of the particle, vo, isresolved into two components, i.e. onealong the OX axis and the other alongthe OY axis, as shown below.

lr= x

vv = j'

Likewise for the acceleratio,n of the particle :

o*= i

A-.: iyJ

al

"o='1 *utr

o?: o1 *

Page 3: Curvilinear Motion and Polar Coordinates

Polar coordinates

The displacement of the particle isexpressed in terms of the radius r and

the angle 0 measured from the OX axis,as shown on the right.

The velocity of the particle is expressed

in terms of a rqdial comPonent and atransverse component, as shown below.

lr=f

ve=r0

)))vi=vi +v-e

Similarly, the acceleration of the particle is expressed

and a transverse component, as shown below.in terms of a radial component

ar= i - '62

oe= r0 + 2i0

azp: a? + a3

Page 4: Curvilinear Motion and Polar Coordinates

Relstionship between Cartesian and porar coordinates

o Displacement:

It can be seen from the figure given on theright that :

x-Y=

r.cos0

r. sin 0

Also:

And :

f=

e- :Vtan 'z-

x

o Velocity:

The velocity of particle p, ( v o ), may be resolved into :

o eitherthe Cartesiancomponents: v* and v, , or

o the Polar components : v, and v,

as shown in the figure below :

*'+y'

Page 5: Curvilinear Motion and Polar Coordinates

From the above figure, it can be seen that :

o r, : vr.cosO + vr.sinO = *.cos0 + y.sinO (1)

o ve = vr.cos0 - vr.sin0 = y.cos0 - *.sinO (2)

Now, from displacement :

x = r.cosO and .lz = r.sin0

* = r.coso - r.sin0.6 (i)

& -tr = r.sin0 + r.cos0.6 Gi)

Therefore, substituting (i) and (ii) into equations (l) and (2) above will produce :

o v, = ( r.cos0 - r.sin0.6 ).cos0 + ( r.sinO + r.cos0.O ).sinO

: r.cos20-rO.sinO.cosO + r.sin2g+rO.cos0.sin0 = i (l):

o v0 = ( r.sin0 + r.cos0.O ).cosO ( r.cosO - r.sin0.0 ).sin0

= r.sinO.cosO + r6.cos2e r.cosO.sinO + rO.sin20 = rd e)

o Acceleration:

The acceleration of particle P, (ap), can be resolved into :

" eitherthe Cartesiancomponents: o, and a, , or

o the Polar components : a, antd au

The diagram for these acceleration components will be identical to the diagram for thevelocities above. Therefore, it will be seen that :

o a, = ar.cosO + ar.sinO = i.cosO + /.sin0 (3)

o ao = ar.cosO - ar.sinO = /.cos0 - x.sinO (4)

Now differentiating (i) and (ii) above will give :

I = r.cosO - r.sin0.6 - r.sin0.6 - r.cos0.02 - r.sin0.6 (iii)

& i = r.sin0+r.cos0.6 + r.cos0.O-r.sin0.02 +r.cos0.6 (iv)

Page 6: Curvilinear Motion and Polar Coordinates

Substituting (iii) and (iv) into equations (3) and (a) for accelerations, and simpliSing,will produce :

o o,=i;-162 (3)

o ao=16+2i6 g)

Summary

The polar components for the velocity and acceleration in curvilinear motion may besummarised as shown below :

o

o

lr=r

vo=,6o ar=i-162

as= 16 +2i6o

Special Cases

o Linear Motion:

When the motion is linear, angle 0 remains constant, giving 0 = 0 and 6 = 0. Hencethe equations above will reduce to :

o Circular Motion:

When the motion is circular, radius r remains constant, giving r = 0 and i = 0. Hencethe equations above will reduce to :

o

o

vr=0

vo = ,6o

o

o, = -'6'ae = rd

or - ra2

or ra

ovr=r

o Vg=0

ar=f

ae=0

o

o