1

Chapter 2Kinematics of Particles

Motions and Coordinates• Motion

– Constrained motion– Unconstrained motion

• Coordinates– Used to describe the motion of particles

2

Motion

Rectilinear motion (1-D)

Plane curvilinear motion (2-D)

Space curvilinear motion (3-D)

CoordinatesRectangular (Cartesian)

coordinates

Normal and tangential coordinates

Polar coordinates

Cylindrical coordinates

Spherical coordinates

),,(),,( zyxyx

)( tn −

),( θr

),,( zr θ

),,( φθr

3

Chapter 2-2. Rectilinear Motion

2

2

Instantaneous velocity

Instantaneous accelerat

:

:

io

n

dsv sdt

dva vdtd v sdt

= =

= =

= =

&

&

&&

vdv adssds sds

=⎧⇒ ⎨ =⎩ & & &&

Graphical Interpretations

4

2 22 1

1 ( ) (the area)2

ads vdv v v= ⇒ − =

a dvv ds=

High School Physics

0

0

2 20 0

20

0 0 0 0

0 0

Given =constant (and ( ) , ( ) , when 0)

(1).

(2).

(3).

2

1

( )

2

v v

a s t s v t v t

dvadt

vdv ads

dsv v a

at

t

v v a s s

s s v atdt

t

= = =

= ⇒

= ⇒

= + =

= +

= + −

= +⇒ +

5

when a≠constant

0

0

0

2 20

00

( ) (1).

(2).

(3).

( ) (1). ( )

( )

1 (

)2

t

os

s

t

v t

v o

dva f t adt

vdv ads

dsvdtd dv dva f v a f vd

v v adt

v v ads

s s v

t tf

d

t v

t

= ⇒ = ⇒

= ⇒

= ⇒

= +

−

= =

=

+

= ⇒ = =

=

⇒ ∫

∫

∫∫

∫

0

0

0 0

0

2 20

0 (2).

( ) (2). 2 ( )

( )

(3).

(

( )

)

v

s t

s

s

s

s

s

v

vdv ds

v v f s

vdv a

ds dt tg s

ds

a f s vdv ads

dsvdt

s s

v s

f

d

v

s

g

= == ⇒

= ⇒ = ⇒

= ⇒= =

+

⇒ =

=

−

∫

∫

∫

∫

∫

Sample 2.1

3

1 1

2 2

( ) 2 24 6

(1). ( ) 72, ?(2). ( ) 30, ( ) ?(3). (4) (3) ?

s t t t

v t tv t a ts s

= − +

= == =− =

6

Problem 2/19

• Small balls fall from rest through the opening at the steady rate of 2 per-second. Find the vertical displacement h of 2 consecutive balls when the lower one has dropped 3 m.

Problem 2/50

• A bumper provides a deceleration as shown in the figure. Suppose a train is approaching the bumper at speed of 40 ft/sec.

• Determine the maximum compression of the bumper.

7

Chapter 2-3. Plane Curvilinear Motion1. 2-D motion: .2. Define the position vector measured from a fixed point .

3. Time derivative of a position vector:

a special case of 3-D

,

rdr dvv r a

t

O

vd dt

= = = =

r

r rr r r r& &

Three coordinates systems to describe the curvilinear motion

Rectangular (Cartesian) coordinates

Normal and tangential coordinates

Polar coordinates

( , )x y

)( tn −

),( θr

8

Chapter 2.4 Rectangular coordinates (x-y)

Vector representation

r xi yj

v r xi yj

a v r xi yj

= +

= = +

= = = +

r rr

r rr r& & &r rr r r& && && &&

0, -Projectile motion: a a gx y= =

9

Sample 2.5

2

( ) 50 16 ,

( ) 100 4 .(0) 0, in meter and in second.

Question :when ( ) 0, ? and ?

xv t t

y t tx y t

y t a v

= −

= −=

= = =

Determine such that is maximized. Rθ

10

Chapter 2.5Normal and Tangential Coordinates (n-t)

• The positive direction of n is always taken toward the center of curvature of the path.

2

2

,

t t t

t t t n

t n

ds ddsv ve e edt

a ve ve ve e

vve e

ρ β

ρβ

ρβ

ρ

=

⇒ = = =

⇒ = + = +

= +

r r r r&

r r r r r& && &

r r&

11

A special case: Circular Motion

22

n

t

v r

va v rr

a r

θ

θ θ

θ

=

= = =

=

&

& &

&&

Write the vector expression for the acceleration of themass G of the simple pendulum in both - and - coordinates for the instance when

o602.00 rad/sec

24.025 rad/sec

a

n t x y

θθθ

==

=

&

&&

12

Exercise 2/119

2 3

A particle moving in the - plane has the position vector as:3 2 ( , ) (in)2 3

Calculate the radius of the path for the position when =2 sec.

x y

P t t

t

=

Chapter 2.6 Polar coordinates (r-θ)

relative to a fixed point

rerrrr

=

13

.2

,

)2()(

,

θ

θ

θθθ

θ

errerra

ererv

err

r

r

r

r&&&&r&&&r

r&r&

r

rr

++−=

+=

=

Sample 2/9

3 2( ) 0.2 0.02 , ( ) 0.2 0.04 .

(3) ? (3) ?

t t t r t t

v a

θ = + = +

= =

14

Exercise 2/145 (slider)

2( ) 0.8 0.05 , ( ) 1.6 0.2 .

(4) ? (4) ? and direction (relative to -axis)

t t t r t t

v a x

θ = − = −

= =

Constant speed =0.6 m/s =1.2 m

, , , , , ?

when 2(1 )3

vR

r r r

t

θ θ θπ=

= +

& &&& &&

15

Chapter 2.7 Space Curvilinear Motion

• Rectangular (x-y-z)• Cylindrical (r-θ-z)• Spherical (R-θ-ψ)

• * n-t coordinates

Rectangular coordinates (x-y-z)

kzjyixRva

kzjyixRv

kzjyixR

r&&

r&&

r&&

&&r&r

r&

r&

r&

&rr

rrrr

++===

++==

++=

16

Cylindrical Coordinates (r-θ-z)

kzerrerrRva

kzererRv

kzerR

r

r

r

r&&

r&&&&r&&&&&r&r

r&

r&r&

&rr

rrr

+++−===

++==

+=

θ

θ

θθθ

θ

)2()( 2

Spherical Coordinates(R-θ-ψ)

φ

θ

φθ

φφθφφ

φθφφφφθ

φθφ

φφθ

eRRR

eRRR

eRRRRva

eReRRv

R

r

r

R

r&&&&

r&&&&&&

r&&&&&&r&r

r&r&r&&rr

rr

)cossin2(

)sin2cos2cos(

)cos(

coseR

eR

2

222

+++

−++

−−===

++==

=

17

Sample 2/11

The power screw starts from rest and is

given a rotational speed which increases

uniformly according to .Suppose the lead of the screw (advancement per revolution) is L. Determine the expressi

θ kt

θ

=

&

&

on for the velocity and acceleration of the center of ball A when the screw has turned through one complete revolution from rest.

Exercise 2/169

• The velocity and acceleration of a particle are given by

• Determine the angle between v and a, , and the radius of curvature.

zyxazyxvrrrr

rrrr

513236

−−=+−=

v&

18

Exercise 2/181

.30

tolowered isit whenboom theof end theof onaccelerati and velocity theof magnitudes theCalculate .sec10

rateconstant at the lowered is boom the time,same At the .min2 of rateconstant a at axis vertical

about the turningis and ,24 lengthof booma has crane revolving The

o

rad/.

rev/

m

Chapter 2.8 Relative Motion

/ ,

/ ,

/ ,

A B A B

A B A B

A B A B

r r r

v v v

a a a

= +

= +

= +

19

Sample 2/12

o

o

Flight A is moving east at a speed of 800 km/hFlight B is moving northeast

(45 ) at a speed Passengers at flight A observe that flight B moves northwest

(60 ).Determine ?

v

v =

Exercise 2/188

cars? theof velocities theconcerning said be can

What cars. theofvelocity relative theof magnitude

theequals cars thebetween distance theof increase of rate time theIf roads.straight along

moving are Band A cars Two

20

Exercise 2/194

• A ship is capable of 16 knots through still water is to maintain a true course due west while encountering a 3-knots current running from north to south. What should be the heading of the ship (measured clockwise from the north to the nearest degree)? How long does it take the ship to proceed 24 nautical miles due west?

Chapter 2.9 Constrained Motion

0202

2

=+=+=+

yxyx

kyx

&&&&

&&

21

02020202

22

2

1

=+−=+=+−=+=+−=+

cDB

DA

cDB

DA

cDB

DA

yyyyyyyyyy

kyyykyy

&&&&&&

&&&&

&&&

&&

Sample 2/15

The tractor is used to hoist the bale with the pulley arrangement shown. If has a forward velocity , determine an expression for the upward velocity of the bale in termsof .

A

B

AB

AV

Vx

22

Exercise 2/207

If block has a leftward velocity of 1.2 m/s, determine the velocityof cylinder .

B

A

Exercise 2/220

The particle is mounted on a loght rod pivoted at andtherefore is constrained in a curvelar arc of radius . Determine the velocity of in terms of the downward velocity of the counterweight fo

B

A O

rA

v r any angle .θ

23

Chapter Review

• Motion– Rectilinear motion (1-D)– Plane curvilinear motion (2-D)– Space curvilinear motion (3-D)

• Coordinates– Rectangular (Cartesian) coordinates

– Normal and tangential coordinates

– Polar coordinates

– Cylindrical coordinates

– Spherical coordinates

),,(),,( zyxyx

)( tn −

),( θr

),,( zr θ

),,( φθr