1
KinematicsKinematics
Course Virtual Worlds
INFOVW  2010
What is Kinematics?What is Kinematics?
• Kinematics is the study of the motion of bodies
without regard to the forces acting on the body
• Focus:
P iti
–
P
os
iti
on
–Velocity
–Acceleration
• How are these related and how do they change
over time?
FocusFocus
• We will focus on
–Particles
–Rigid bodies
•
A
syste
m
o
f
pa
r
t
i
c
l
es
t
h
at
r
e
m
a
in
at
fix
ed
d
i
sta
n
ces
syste o pa t c es t at e a at ed d sta ces
from each other with no relative translation or
rotation among them
Velocity and AccelerationVelocity and Acceleration
2
• Velocity is a vector quantity,
–Direction
–Magnitude
d f l ll d d
SpeedSpeed
• Magnitu
d
e o
f
ve
l
ocity is ca
ll
e
d
spee
d
t
s
v
Speed exampleSpeed example
1000 m
kph
t
s
v 72
s
m
20
s)1060(
m1000
1
t
1
= 10 s
2
t
2
= 60 s
Instantaneous velocityInstantaneous velocity
ds
s
dt
ds
t
s
v
t
)(lim
0
Integrate this…
Relationship between s and v
Relationship between s and v
dtvds
dt
ds
v
2
1
2
1
2
1
12
t
t
t
t
s
s
dtvsss
dtvds
3
Displacement and DistanceDisplacement and Distance
• In 1D (straightline movement) displacement
and distance traveled is the same
•
I hi h di i thi i diff t
•
I
n
hi
g
h
er
di
mens
i
ons
thi
s
i
s
diff
eren
t
Show this on whiteboard…
AccelerationAcceleration
t
v
a
dt
dv
a
t
v
a
t
0
lim
Integrate this…
Speed change…
Speed change…
dtadv
dt
dv
a
2
1
2
1
2
1
12
t
t
v
v
t
t
dtavvv
dtadv
Constant Acceleration
Constant Acceleration
4
Constant AccelerationConstant Acceleration
• If an object experiences constant acceleration its
speed changes accordingly
–Example is the acceleration due to the earth’s
gravity (a = g = 9 81 m/s2)
gravity
(a
=
g
=
9
.
81
m/s2)
• This can be calculated by solving the following:
2
1
2
1
v
v
t
t
dtadv
Do this on whiteboard…
Velocity dependent on distance
Velocity dependent on distance
d
ds
d
dt
dv
a
dvvdsa
dv
dt
d
sa
Kinematic Diffential Equation of Motion
Integrate this…
ExamExam
• You should be able to calculate all these things
by heart
• Table 2.1 in the book will not be supplied at
the exam
NonNonConstant AccelerationConstant Acceleration
5
NonNonconstant accelerationconstant acceleration
• This is very common
• For example, any object moving in a real world
will experience drag
d i l t l t
–more on
d
rag
i
n
l
a
t
er
l
ec
t
ure
• One type of drag is dependent on speed
2
vka
Resulting equation…
Resulting equation…
k
dv
vka
2
2
dtkdv
v
v
k
dt
2
2
1
Integrate this…
…integrated……integrated…
1
2
dtkdv
v
)1(
...
1
1
2
tkv
v
v
…substitute……substitute…
dsdtv
dt
ds
v
)1(
where
1
1
tkv
v
v
Integrate this…
6
…results in…results in
t
k
v
)
1ln
(
1
k
s
)
(
1
Question for you
Question for you
• When and where will an object stop under the
drag in the previous example?
•
U th f ll i l
•
U
se
th
e
f
o
ll
ow
i
ng va
l
ues
–k = 10
–v1 = 20 m/s
General casesGeneral cases
• In general very hard to calculate using these
formulas
• Usually solved by numerical integration
•
Will b di d i l t l t
•
Will
b
e
di
scusse
d
i
n
l
a
t
er a
l
ec
t
ure
2D Particle Kinematics2D Particle Kinematics
7
Independence of two directionsIndependence of two directions
• In the 2D case, you can regard the two
directions as being independent
–Two sets of 1D problems
y
x
y
x
a
a
v
v
y
x
a
v
s
Written out this leaves…Written out this leaves…
y
x
v
v
d
dy
dt
dx
dt
y
x
d
dt
ds
v
y
x
a
a
dt
yd
dt
xd
dt
d
dt
d
d
t
2
2
2
2
2
2
sv
a
An ExampleAn Example
v
1
=800 m/s
??
30°
assume: g = 10 m/s
2
3D Particle Kinematics3D Particle Kinematics
8
Just an extension of 2DJust an extension of 2D
• Nothing more complicated than 2D case
• Just add an extra dimension
•
R lt i
•
R
esu
lt
s
i
n:
–position: x, y, z
–velocity: v
x
, v
y
, v
z
–acceleration: a
x
, a
y
, a
z
Rigid Body KinematicsRigid Body Kinematics
Similar Kinematics
Similar Kinematics
• Rigid Body Kinematics is basically particle
kinematics with rotation
•
M t i t i t th t f
•
M
os
t
conven
i
en
t
i
s
t
o use
th
e cen
t
er o
f
mass as
the particle for linear kinematics
–track C.o.M. translation
–track rotation around C.o.M
Local Coordinate FrameLocal Coordinate Frame
Ω
body frame
y
y
x
Ω
world frame
x
9
Angular Velocity and AccelerationAngular Velocity and Acceleration
dt
d
dd
dt
d
dt
d
2
2
Integrate this…
Points on the ObjectPoints on the Object
• Points on the object move
• Combination of two motions:
–linear motion of CoM
angular motion around CoM
–
angular
motion
around
CoM
• Want to calculate this because you want to
know stuff about the points
–For example, how hard will two object hit each
other
Arc LengthArc Length
• Call c
p
the arc length for a point on the object
• Let r
p
be the distance between this point and
the axis of rotation
L
Ω
扨 汨 b d (
L
整e
Ω
b
攠e
h
攠慮e
l
攠e
h
攠e
b
橥捴潴慴j
d
(
楮i
牡摩慮猩
pp
rc
Angular and Linear Velocity
Angular and Linear Velocity
p
dt
d
r
dt
dc
pp
rv
Differentiate this…
10
Angular and Linear AccelerationAngular and Linear Acceleration
dt
d
r
dt
dv
p
p
ra
t
Tangential linear acceleration
Centripetal Acceleration
Centripetal Acceleration
• Besides the tangential linear acceleration, there
is also the centripetal acceleration of a point on
the object
•
Thi i di t d t d th i f t ti
•
Thi
s
i
s
di
rec
t
e
d
t
owar
d
th
e ax
i
s o
f
ro
t
a
ti
on
• This is what you ‘feel’ when you go through a
corner in a car or bus
Centripetal AccelerationCentripetal Acceleration
2
r
v
a
n
2
ra
r
n
2D vs. 3D
2D vs. 3D
• In 2D there is no problem in using these scalar
quantities for angular speed and acceleration
•
H i 3D thi i f bl d
•
H
owever,
i
n
3D
thi
s
i
s more o
f
a pro
bl
em, an
d
vectors need to be used
11
Linear Tangential VelocityLinear Tangential Velocity
r
ωv
rαa
rωωa
t
n
)(
Resulting Quantities for Point
Resulting Quantities for Point
• Remember, the object moves linearly as the
CoM moves
• Rotation add to the movement for points on
th bj t
th
e o
bj
ec
t
• Total motion of a point on the object is the
sum of the two motions
Show on whiteboard…
ExerciseExercise
• Car drives around a bend with 20 m/s
• The diameter of the turn is 20 meter.
• Questions:
1
Wh t i th l l ti?
1
.
Wh
a
t
i
s
th
e angu
l
ar acce
l
era
ti
on
?
2.What is the centripetal acceleration?
3.Assume that the gravitational acceleration is 10
m/s
2
, what is the amount of Gs experienced
by the driver and passengers?
Questions??Questions??
12
Next Lecture…Next Lecture…
• Topic of the next lecture:
–Forces
Enter the password to open this PDF file:
File name:

File size:

Title:

Author:

Subject:

Keywords:

Creation Date:

Modification Date:

Creator:

PDF Producer:

PDF Version:

Page Count:

Preparing document for printing…
0%
Comments 0
Log in to post a comment