Curvilinear Motion

Lecture III

Topics Covered in Curvilinear Motion Plane curvilinear motion Coordinates used for describing

curvilinear motion Rectangular coords n-t coords Polar coords

Plane curvilinear Motion Studying the motion of a particle along a

curved path which lies in a single plane (2D). This is a special case of the more general 3D

motion.

3D

Plane curvilinear Motion – (Cont.)

If the x-y plane is considered as the plane of motion; from the 3D case, z and are both zero, and R becomes as same as r.

The vast majority of the motion of particles encountered in engineering practice can be represented as plane motion.

Coordinates Used for Describing the Plane Curvilinear Motion

Rectangular coordinates

Normal-Tangential coordinates

Polar coordinate

s

y

x

P

t

nPA

PB

PC

t

n

t

n

y

x

Pr

r

Path

Path

Path

O O

Plane Curvilinear Motion – without Specifying any Coordinates

(Displacement)

Note: Since, here, the particle motion is described by two coordinates components, both the magnitude and the direction of the position, the velocity, and the acceleration have to be specified.

P at time t

P at time t+t r

r(t)

r(t+t)

O

or r(t)+r(t)

s

t(

Note: If the origin (O) is changed to some different location, the position r(t) will

be changed, but r(t) will not change.

Actual distance traveled by the particle (it is s scalar)

The vector displacement of the particle

Plane Curvilinear Motion – without Specifying any Coordinates (Velocity)

Average velocity (vav):

Instantaneous velocity (v): as t approaches zero in the limit,

t

r

vav

Note: vav has the direction of r and its magnitude equal to the magnitude of r divided by t.Note: the average speed of the particle is the scalar s/t. The magnitude of the speed and vav approach one another as t approaches zero.

rrr

v lim0

dt

d

tt

Note: the magnitude of v is called the speed, i.e.

v=|v|=ds/dt= s..

v(t)

v(t)

v(t+t)

v(t+t) v

P

P

Note: the velocity vector v is always tangent to the path.

Plane Curvilinear Motion – without Specifying any Coordinates

(Acceleration)

Average Acceleration (aav):

Instantaneous Acceleration (a): as t approaches zero in the limit,

t

v

a av

Note: aav has the direction of v and its magnitude is the magnitude of v divided by t.

rvvv

a lim0

dt

d

tt

Note: in general, the acceleration vector a is neither tangent nor normal to the path. However, a is tangent to the hodograph.

C Hodograph

P

P

V1

V2

V1

V2

a1a2

The description of the Plane Curvilinear Motion

in the Rectangular Coordinates (Cartesian

Coordinates)

Plane Curvilinear Motion - Rectangular Coordinates

jijirva :on vectoraccelerati The

jijirv :vector velocity The

jir :ectorposition v The

yxaa

yxvv

yx

yx

yx

y

x

PPath

O

j

i

r

v

vy

vx

aay

ax P

22

22

22

:on vectoraccelerati theof magnitude The

:vector velocity theof magnitude The

:ectorposition v theof magnitude The

yx

yx

aaa

vvv

yxr

Note: the time derivatives of the unit vectors are zero because their magnitude and direction remain constant.

Note: if the angle is measured counterclockwise from the x-axis to v for the configuration of the axes shown, then we can also observe that dy/dx = tan = vy/vx.

Plane Curvilinear Motion - Rectangular Coordinates (Cont.)

The coordinates x and y are known independently as functions of time t; i.e. x = f1(t) and y = f2(t). Then for any value of time we can combine them to obtain r.

Similarly, for the velocity v and for the acceleration a. If is a given, we integrate to get v and integrate again

to get r. The equation of the curved path can obtained by

eliminating the time between x = f1(t) and y = f2(t). Hence, the rectangular coordinate representation of

curvilinear motion is merely the superposition of the components of two simultaneous rectilinear motions in x- and y- directions.

Plane Curvilinear Motion - Rectangular Coordinates (Cont.) –

Projectile Motion

)(2

)21(

)( )(

0

22y

2

ooy

oyooxo

oyyoxx

yx

yyg)(vv

gtt)(vyyt )(vxx

gtvvvv

gaa

x

Path

y

vo

(vx)o = vo cos

(vy)o = vo sing

vvy

vx

vvy

vx

Exercises

Exercise # 1

Given the position vector , calculate the velocity, acceleration, and the magnitudes of position, velocity and acceleration.

j2ir 32 tt

Exercise # 2

2/62: A particle which moves with curvilinear motion has coordinates in millimeters which vary with the time t in seconds according to x = 2t2 - 4t and y = 3t2 – (1/3)t3. Determine the magnitudes of the velocity v and acceleration a and the angles which these vectors make with the x-axis when t = 2 s.

Exercise # 32/63: A roofer tosses a small tool toward a coworker on the ground. What is the minimum horizontal velocity vo necessary so that the tool clears point B? Locate the point of impact by specifying the distance s shown in the figure.

Exercise # 42/68: A rocket runs out of fuel in the position shown and continues in unpowered flight above the atmosphere. If its velocity in this position was 1000 km/h, calculate the maximum additional altitude h acquired and the corresponding time t to reach it. The gravitational acceleration during this phase of its flight is 9.39 m/s2.

Exercise # 52/80: The pilot of an airplane carrying a package of mail to a remote outpost wishes to release the package at the right moment to hit the recovery location A. What angle with the horizontal should the pilot's line of sight to the target make at the instant of release? The airplane is flying horizontally at an altitude of 100 m with a velocity of 200 km/h.

Exercise # 62/92: The basketball player likes to release his foul shots at an angle = 50° to the horizontal as shown. What initial speed vo will cause the ball to pass through the center of the rim?

Exercise # 72/73: A particle is ejected from the tube at A with a velocity v at angle with the vertical y-axis. A strong horizontal wind gives the particle a constant horizontal acceleration a in the x-direction. If the particle strikes the ground at a point directly under its released position, determine the height h of point A. The downward y-acceleration may be taken as the constant g.