Plane curvilinear motion is the motion of - deu.edu.tr

Plane curvilinear motion is the motion of
a particle along a curved path which lies
in a single plane.

Before the description of plane
curvilinear motion in any specific set of
coordinates, we will use vector analysis
to describe the motion, since the results
will be independent of any particular
coordinate system.

AobrytigttihinmeOep.ot BstiohtteiohpntavhreetcicmtloearginsritamutdepeaossauintrdieoddnifrAer,ocwmthioitnchheofifsirxloecdaatreed

kloncoawtnedatbytimtheet.posAittiotnimveectt+oDrt,rtheDrpa.rticle is at A' ,

Path of particle 
V

t+Dt A' A' 
 DV
r  Dr Dr Ds V 
V a
r 
A AV A
t

O

 The displacement of the
V
Path of particle particle d uring Dt is the
r which represents
t+Dt A' vector D

Dr Ds A'  the vector change of
 DV
r  Dr V  position and is independent
V
r  a of the choice of origin. If

A AV another point was selected
t
A as the origin the position

O bveucttDorrs would have changed
would remain the

same.

Path of particle 
V

t+Dt A' A' 
 DV
r  Dr Dr Ds V 
V a
r 
A AV A
t

O

The distance actually travelled by the particle as it

moves along the path from A to A' is the scalar

length Ds measured along the pDatrh. . It is important
to distinguish between Ds and

Velocity

The average velocity dDoefrfitnheed particle

between A and A' is as

  Dr
vav Dt

owfhDicrh .isTahevemcatgonritwuhdoeseofdivraevcitsionDirs that
.

Dt

The average speed of the particle

between A and A' is

Ds

Dt

Clearly, thDer magnitude of the average
velocity Ds
Dt and the speed Dt approach

one another as the interval Dt decreases

and A and A' become closer together.

The instantaneous velocity v of the particle is defined as the

limiting value of the average velocity as the time Dt approaches

zero. v Dr dr r
Dt dt
 lim  

Dt 0


We observe that the direction of v approaches that of the

tangent to the path as Dt approaches zero and, thus, the

velocity is always a vector tangent to the path.

The magnitude of v is called the speed and is the scalar

v  v  ds  s
dt

The acrheanvgeatinAvealnodcitviesa, twAhi‘cdhurairnegttainmgeenDtt to the path Dv .
and is a vector

v Dv

v

Here Dv indicates both change in magnitude and direction
of v . Therefore, when the differential of a vector is to

be taken, the changes both in magnitude and direction
must be taken into account.

Acceleration

The average acceleration of the particle between

A and A' is defined as Dv
Dt
 
aav

wofhicDhv is a vector whose dDivrection is that
. Its magnitude is

Dt

The instantaneous acceleration of the particle is
defined as the limiting value of the average
acceleration as the time interval approaches zero.

a  lim Dv  dv  v  r
Dt dt
Dt 0

dAisreDcttbioencoomf eDsvsmapapllreoraacnhdesapdpvro. aches zero, the

The acceleration includes the effects of both the
changes in magnitude and direction of v . In

general, the direction of the acceleration of a
particle in curvilinear motion is neither tangent to
the path nor normal to the path.

If the acceleration was divided
into two components one tangent
and the other normal to the
path, it would be seen that the
normal component would always
be directed towards the center
of curvature.

If velocity vectors are plotted from some arbitrary
point C, a curve, called the hodograph, is formed.
Acceleration vectors are tangent to the hodograph.

Three different coordinate systems are commonly
used in describing the vector relationships for plane
curvilinear motion of a particle. These are:

• Rectangular (Cartesian) Coordinates
(Kartezyen veya Dik Koordinatlar)

• Normal and Tangential Coordinates
(Doğal veya Normal-Teğetsel Koordinatlar)

• Polar Coordinates
(Polar veya Kutupsal Koordinatlar)

The selection of the appropriate reference system is a
prerequisite for the solution of a problem. This
selection is carried out by considering the description
of the problem and the manner the data are given.

Cartesian Coordinate system is useful for describing motions
where the x- and y-components of acceleration are
independently generated or determined. Position, velocity and
acceleration vectors of the curvilinear motion are indicated by
their x and y components.

v y a 
ay
j
Path of particle
vy

 A q axa A
ryj

vx

 x
O xi i

Let us assume that at time t the particle is at point A.
With the aid of the unit vectors i and j, we can write

the position, velocity and acceleration vectors in terms

of x- and y-comrponxeintsy. j
v  
a  v i  vy j xi  yj
 x    xi yj
 j
axi  ay j  vxi  vy  

v y a  As we differentiate with
ay respect to time, we
j
Path of particle observe that the time
vy derivatives of the unit

 A q a
ryj ax
vx A vectors are zero because

  their magnitudes and
O xi i
x directions remain constant.

The magnitudes of the components of v and a are:

vx  x vy  y
ax  vx  x ay  vy  y

In the figure it is seen that the direction of ax is
in –x direction. Therefore when writing in vector

form a “-” sign must be added in front of ax.

v y a 
ay
j
Path of particle
vy

 A q a
ryj ax A

vx

 x
O xi i

The direction of the velocity is always tangent to the
path. No such thing can be said for acceleration.

v2  vx2  vy2
v  vx2  vy2

tan q  v y

vx
a2  ax2  ay2
a  ax2  ay2

tan a  a y

ax

If the coordinates x and y are known independently

as functions of time, x=f1(t) and y=f2(t), then for any
value of the time we can obtain rax.nSadnimdyilayrtlotyo,owbobteatianinav.
combine their first derivatives

and their second derivatives x

Inversely, if ax and ay are known, then we must take
integrals in order to obtain the components of
velocity and position. If time t is removed between x
and y, the equation of the path can be obtained as

y=f(x).

Projectile Motion (Eğik Atış Hareketi)

An important application of two-dimensional kinematic theory is the
problem of projectile motion. For a first treatment, we neglect
aerodynamic drag and the curvature and rotation of the earth, and we
assume that the altitude change is small enough so that the acceleration
due to the gravity can be considered constant. With these assumptions,
rectangular coordinates are useful to employ for projectile motion.

Acceleration components;

ax=0

ay= -g

y Apex; vy=0

vy v vx

voy= vosinq vo vx g vx
v'
v'y

q x
vox= vocosq

If motion is examined separately in horizontal and
vertical directions,

Horizontal Vertical

ax  0 ay  -g
vx  constant
vy  v0 y - gt
vx  v0x
y  y0  v0 y t - 1 gt 2
x  x0  v0xt 2

vy2  v0y2 - 2g( y - y0 )

We can see that the x- and y-motions are
independent of each other. Elimination of the time t
between x- and y-displacement equations shows the
path to be parabolic.

1. The particle P moves along the curved slot, a

portion of which is shown. Its distance in meters

measured along the slot is given by s=t2/4, where t is

in seconds. The particle is at A when t = 2.00 s and

at B when t =2.20 s. Determine the magnitude aav of

the average acceleration of P between AaavanudsinBg. Also
a vector unit
express ithanedacj c.eleration as
vectors

2. With what minimum horizontal velocity u can a
boy throw a rock at A and have it just clear the
obstruction at B?

3. For a certain interval of motion, the pin P is

forced to move in the fixed parabolic slot by the

vertical slotted guide, which moves in the x direction

at the constant rCaatlceuolafte20thmemm/asg.nAitlul dmeesaosfurvemanedntas
are in mm and s.

of pin P when x = 60 mm.

4. A projectile is fired with a velocity u at right

angles to the slope, which is inclined at an angle q

with the horizontal. Derive an expression for the
distance R to the point of impact.

5. Pins A and B must always remain in the vertical
slot of yoke C, which moves to the right at a
constant speed of 6 cm/s. Furthermore, the pins
cannot leave the elliptic slot. What is the speed at
which the pins approach each other when the yoke
slot is at x = 50 cm? What is the rate of change of
speed toward each other when the yoke slot is again
at x = 50 cm?

60 cm yoke
100 cm 6 cm/s

x