Linear and Rotational Motion in Physics
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Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
1
PHYS 1443 – Section 003
Lecture #18
Monday, April 19, 2021
Dr.
Jae
hoon
Yu
•
CH7
:
L
inear
M
omentum
•
Motion
o
f
a
G
roup
o
f
O
bjects
•
CH8
:
R
otational
M
otion
•
Fundamentals
o
f
R
otational
M
otion
•
Rotational
K
inematics
•
Torque
&
V
ector
P
roduct
•
Moment
o
f
I
nertia
Today’s
homework
is
homework
#
10
,
due
11pm
,
Tuesday
,
May
4
!!
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
2
Announcements
•
The
s
kipped
h
omework
#
9
i
s
d
ue
1
1pm
,
T
uesday
,
A
pril
2
7
!
!
•
Special
p
roject
#
6
d
eadline
e
xtended
t
o
n
ext
M
onday
,
A
pr
.
2
6
•
2
nd
n
on-comprehensive
e
xam
r
esults
–
Class
a
verage
:
6
3.3/102
•
Equivalent
t
o
6
2/100
•
Previous
r
esults
:
7
7.3/100
a
nd
5
0.2/100
–
Top
s
core
:
9
6/102
•
Evaluation
P
olicy
–
Homework
:
2
5
%
–
Final
e
xam
:
2
3
%
–
Mid-term
e
xam
:
2
0
%
–
One
b
etter
o
f
t
he
t
wo
t
erm
e
xams
:
1
2
%
–
Lab
:
1
0
%
–
Quizzes
:
1
0
%
–
Extra
c
redit
:
1
0
%
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
3
The
net
effect
of
these
small
gravitational
forces
is
equivalent
to
a
single
force
acting
on
a
point
(
Center
of
Gravity
)
with
mass
M
.
Center of Mass and Center of Gravity
The
center
of
mass
of
any
symmetric
object
lies
on
the
axis
of
symmetry
and
on
any
plane
of
symmetry
,
if
the
object’s
mass
is
evenly
distributed
throughout
the
body
.
Center
of
Gravity
How
do
you
think
you
can
determine
the
CM
of
the
objects
that
are
not
symmetric
?
One
can
use
gravity
to
locate
CM
.
1.
Hang
t
he
o
bject
b
y
o
ne
p
oint
a
nd
d
raw
a
v
ertical
l
ine
f
ollowing
a
p
lum-bob
.
2.
Hang
t
he
o
bject
b
y
a
nother
p
oint
a
nd
d
o
t
he
s
ame
.
3.
The
p
oint
w
here
t
he
t
wo
l
ines
m
eet
i
s
t
he
C
M
.
Since
a
rigid
object
can
be
considered
as
a
collection
of
small
masses
,
one
can
see
the
total
gravitational
force
exerted
on
the
object
as
What
does
this
equation
tell
you
?
The
CoG
is
the
point
in
an
object
as
if
all
the
gravitational
force
is
acting
on
!
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
4
Motion of a Group of Particles
We’ve
learned
that
the
CM
of
a
system
can
represent
the
motion
of
a
system
.
Therefore
,
for
an
isolated
system
of
many
particles
in
which
the
total
mass
M
is
preserved
,
the
velocity
,
total
momentum
,
acceleration
of
the
system
are
Velocity
of
the
system
Total
Momentum
of
the
system
Acceleration
of
the
system
The
external
force
acting
on
the
system
If
net
external
force
is
0
System’s
momentum
is
conserved
.
What
about
the
internal
forces
?
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
5
In
the
simplest
kind
of
rotation
,
all
points
on
a
rigid
object
move
on
circular
paths
around
an
axis
of
rotation
.
Rotational Motion and Angular Displacement
The
angle
swept
out
by
the
line
passing
through
any
point
on
the
body
and
intersecting
the
axis
of
rotation
perpendicularly
is
called
the
angular
displacement
.
It’s
a
vector
!!
So
,
there
must
be
a
direction
…
(
poll
6
)
How
do
we
define
directions
?
+:
if
counter-clockwise
-:
if
clockwise
The
direction
vector
points
gets
determined
based
on
the
right-hand
rule
.
These
are
just
conventions
!!
For
one
full revolution:
SI Unit of the Angular Displacement
Since
the
circumference
of
a
circle
is
2
r
Dimension?
None
O
n
e
r
a
d
i
a
n
i
s
a
n
a
n
g
l
e
s
u
b
t
e
n
d
e
d
b
y
a
n
a
r
c
o
f
t
h
e
s
a
m
e
l
e
n
g
t
h
a
s
t
h
e
r
a
d
i
u
s
!
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
6
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
7
Unit of the Angular Displacement
1
radian
is
And
one
degrees
is
How
many
degrees
are
in
one
radian
?
How
radians
is
one
degree
?
How
many
radians
are
in
10.5
revolutions
?
Very
important
:
In
solving
angular
problems
,
all
units
,
degrees
or
revolutions
,
must
be
converted
to
radians
!
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
8
Example
A
particular
bird
’
s
eyes
can
distinguish
objects
that
subtend
an
angle
no
smaller
than
about
3x10
-4
rad
.
(
a
)
How
many
degrees
is
this
?
(
b
)
How
small
an
object
can
the
bird
just
distinguish
when
flying
at
a
height
of
100m
?
(a) One radian is 360
o
/2
. Thus
(b) Since l=r
and for small angle
arc length is approximately the
same as the chord length.
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
9
Synchronous
satellites
are
put
into
an
orbit
whose
radius
is
4.23×10
7
m
.
If
the
angular
separation
of
the
two
satellites
is
2.00
degrees
,
find
the
arc
length
that
separates
them
.
Ex. Adjacent Synchronous Satellites
Convert
degrees to
radians
What
do
we
need
to
find
out
?
The
Arc
length
!!!
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
10
10
The
diameter
of
the
Sun
is
about
400
times
greater
than
that
of
the
moon
.
By
coincidence
,
the
sun
is
also
about
400
times
farther
from
the
earth
than
is
the
moon
.
For
an
observer
on
the
earth
,
compare
the
angle
subtended
by
the
moon
to
the
angle
subtended
by
the
sun
and
explain
why
this
result
leads
to
a
total
solar
eclipse
.
Ex. A Total Eclipse of the Sun
I
can
even
cover
the
entire
sun
with
my
thumb
!!
Why
?
Because
the
distance
(
r
)
from
my
eyes
to
my
thumb
is
far
shorter
than
that
to
the
sun
.
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
11
11
Using
what
we
have
learned
earlier
,
how
would
you
define
the
angular
displacement
?
Angular Displacement, Velocity, and Acceleration
How
about
the
average
angular
speed
?
And
the
instantaneous
angular
speed
?
By
the
same
token
,
the
average
angular
acceleration
is
defined
as
…
And
the
instantaneous
angular
acceleration
?
When
rotating
about
a
fixed
axis
,
every
particle
on
a
rigid
object
rotates
through
the
same
angle
and
has
the
same
angular
speed
and
angular
acceleration
.
Unit
?
rad
/
s
Unit
?
rad
/
s
Unit
?
rad
/
s
2
Unit
?
rad
/
s
2
Dimension
(
poll
5
)?
[
T
-1
]
Dimension
(
poll
5
)?
[
T
-1
]
Dimension
(
poll
5
)
?
[
T
-2
]
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
12
12
Rotational Kinematics
The
first
type
of
motion
we
have
learned
in
linear
kinematics
was
under
the
constant
acceleration
.
We
will
learn
about
the
rotational
motion
under
constant
angular
acceleration
,
because
these
are
the
simplest
motions
in
both
cases
.
Just
like
the
case
in
linear
motion
,
one
can
obtain
Angular
velocity
under
constant
angular
acceleration
:
Angular
displacement
under
constant
angular
acceleration
:
One
can
also
obtain
Linear
kinematics
Linear
kinematics
Linear
kinematics
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
13
13
Problem Solving Strategy
•
Visualize
t
he
p
roblem
b
y
d
rawing
a
p
icture
.
•
Write
d
own
t
he
v
alues
t
hat
a
re
g
iven
f
or
a
ny
o
f
t
he
f
ive
k
inematic
v
ariables
a
nd
c
onvert
t
hem
t
o
S
I
u
nits
.
–
Remember
t
hat
t
he
u
nit
o
f
t
he
a
ngle
m
ust
b
e
r
adians
!
!
•
Verify
t
hat
t
he
i
nformation
c
ontains
v
alues
f
or
a
t
l
east
t
hree
o
f
t
he
f
ive
k
inematic
v
ariables
.
S
elect
t
he
a
ppropriate
e
quation
.
•
When
t
he
m
otion
i
s
d
ivided
i
nto
s
egments
,
r
emember
t
hat
t
he
f
inal
a
ngular
v
elocity
o
f
o
ne
s
egment
i
s
t
he
i
nitial
v
elocity
f
or
t
he
n
ext
.
•
Keep
i
n
m
ind
t
hat
t
here
m
ay
b
e
t
wo
p
ossible
a
nswers
t
o
a
k
inematics
p
roblem
.
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
14
14
Ex. Rotational Kinematics
A
wheel
rotates
with
a
constant
angular
acceleration
of
3.50
rad
/
s
2
.
If
the
angular
speed
of
the
wheel
is
2.00
rad
/
s
at
t
i
=
0
,
a
)
through
what
angle
does
the
wheel
rotate
in
2.00s
?
Using
the
angular
displacement
formula
in
the
previous
slide
,
one
gets
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
15
15
Example for Rotational Kinematics cnt’d
What
is
the
angular
speed
at
t
=
2.00s
?
Using
the
angular
speed
and
acceleration
relationship
Find
the
angle
through
which
the
wheel
rotates
between
t
=
2.00s
and
t
=
3.00s
.
How
many
revolution
is
it
?
Using
the
angular
kinematic
formula
At
t
=
2.00s
At
t
=
3.00s
Angular
displacement
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
16
16
Relationship Between Angular and Linear Quantities
What
do
we
know
about
a
rigid
object
that
rotates
about
a
fixed
axis
of
rotation
?
When
a
point
rotates
,
it
has
both
the
linear
and
angular
components
in
its
motion
.
What
is
the
linear
component
of
the
motion
you
see
?
Every
particle
(
or
masslet
)
in
the
object
moves
in
a
circle
centered
at
the
same
axis
of
rotation
.
Linear
velocity
along
the
tangential
direction
.
How
do
we
relate
this
linear
component
of
the
motion
with
angular
component
?
The
arc-length
is
So
,
the
tangential
speed
v
is
What
does
this
relationship
tell
you
about
the
tangential
speed
of
the
points
in
the
object
and
their
angular
speed
?:
Although
every
particle
in
the
object
has
the
same
angular
speed
,
its
tangential
speed
differs
and
is
proportional
to
its
distance
from
the
axis
of
rotation
.
The
farther
away
the
particle
is
from
the
center
of
rotation
,
the
higher
the
tangential
speed
.
The
direction
of
ω
follows
the
righthand
rule
.
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
17
17
Is the lion faster than the horse?
A
rotating
carousel
has
one
child
sitting
on
a
horse
near
the
outer
edge
and
another
on
a
lion
halfway
out
from
the
center
.
(
a
)
Which
child
has
the
greater
linear
speed
?
(
b
)
Which
child
has
the
greater
angular
speed
?
(a)
Linear
s
peed
i
s
t
he
d
istance
t
raveled
d
ivided
b
y
t
he
t
ime
i
nterval
.
S
o
,
t
he
c
hild
s
itting
a
t
t
he
o
uter
e
dge
t
ravels
m
ore
d
istance
w
ithin
t
he
g
iven
t
ime
t
han
t
he
c
hild
s
itting
c
loser
t
o
t
he
c
enter
.
T
hus
,
t
he
h
orse
i
s
f
aster
t
han
t
he
l
ion
.
(
b
)
Angular
speed
is
the
angle
traveled
divided
by
the
time
interval
.
The
angle
both
the
children
travel
in
the
given
time
interval
is
the
same
.
Thus
,
both
the
horse
and
the
lion
have
the
same
angular
speed
.
Monday, April 19, 2021
PHYS 1443-003, Spring 2021
Dr. Jaehoon Yu
18
18
How about the acceleration?
Two
How
many
different
linear
acceleration
components
do
you
see
in
a
circular
motion
and
what
are
they
?
Total
linear
acceleration
is
Since
the
tangential
speed
v
is
What
does
this
relationship
tell
you
?
Although
every
particle
in
the
object
has
the
same
angular
acceleration
,
its
tangential
acceleration
differs
proportional
to
its
distance
from
the
axis
of
rotation
.
Tangential
,
a
t
,
and
the
radial
acceleration
,
a
r
.
The
magnitude
of
tangential
acceleration
a
t
is
The
radial
or
centripetal
acceleration
a
r
is
What
does
this
tell
you
?
The
father
away
the
particle
is
from
the
rotation
axis
,
the
more
radial
acceleration
it
receives
.
In
other
words
,
it
receives
more
centripetal
force
.
Explore the concepts of linear momentum, center of mass, rotational motion, and angular displacement in physics. Learn how to determine the center of mass of objects, analyze motion of particle groups, and understand the conservation of momentum in systems under external forces. Delve into the fundamentals of rotational motion, including torque, moment of inertia, and angular kinematics. Discover the importance of symmetry, equilibrium, and gravitational forces in describing the motion of objects.
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PHYS 1443 Section 003 Lecture #18 Monday, April 19, 2021 Dr. Jae Jaehoon Yu CH7: Linear Momentum Motion of a Group of Objects CH8: Rotational Motion Fundamentals of Rotational Motion Rotational Kinematics Torque & Vector Product Moment of Inertia Today s homework is homework #10, due 11pm, Tuesday, May 4!! Yu Monday, April 19, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 1
Announcements The skipped homework #9 is due 11pm, Tuesday, April 27!! Special project #6 deadline extended to next Monday, Apr. 26 2nd non-comprehensive exam results Class average: 63.3/102 Equivalent to 62/100 Previous results: 77.3/100 and 50.2/100 Top score: 96/102 Evaluation Policy Homework: 25% Final exam: 23% Mid-term exam: 20% One better of the two term exams: 12% Lab: 10% Quizzes: 10% Extra credit: 10% Monday, April 19, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 2
Center of Mass and Center of Gravity The center of mass of any symmetric object lies on the axis of symmetry and on any plane of symmetry, if the object s mass is evenly distributed throughout the body. One can use gravity to locate CM. 1. Hang the object by one point and draw a vertical line following a plum-bob. 2. Hang the object by another point and do the same. 3. The point where the two lines meet is the CM. Since a rigid object can be considered as a collection of small masses, one can see the total gravitational force exerted on the object as CM Axis of symmetry How do you think you can determine the CM of the objects that are not symmetric? Center of Gravity mi The net effect of these small gravitational forces is equivalent to a single force acting on a point (Center of Gravity) with mass M. What does this equation tell you? mig g Monday, April 19, 2021 The CoG is the point in an object as if all the gravitational force is acting on! PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 3
Motion of a Group of Particles We ve learned that the CM of a system can represent the motion of a system. Therefore, for an isolated system of many particles in which the total mass M is preserved, the velocity, total momentum, acceleration of the system are i mivi = 1 M midri dt drCM dt 1 M i d dt i i mivi = vCM= = Velocity of the system miri M Total Momentum of the system i = i pCM= = = =ptot MvCM= dvCM dt M mivi pi M i miai = 1 M 1 M d dt dvi dt Acceleration of the system i i aCM= = mivi mi M a = The external force acting on the system What about the internal forces? dptot dt i Ma = = mi Fext dptot dt System s momentum is conserved. = ptot=const 0= Fext If net external force is 0 Monday, April 19, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 4
Rotational Motion and Angular Displacement In the simplest kind of rotation, all points on a rigid object move on circular paths around an axis of rotation. The angle swept out by the line passing through any point on the body and intersecting the axis of rotation perpendicularly is called the displacement. = angular o It s a vector!! So, there must be a direction (poll 6) +:if counter-clockwise -:if clockwise How do we define directions? Monday, April 19, 2021 The direction vector points gets determined based on the right-hand rule. PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 5 These are just conventions!!
SI Unit of the Angular Displacement Arc length Radius =s (in radians) = r Dimension? None For one full revolution: Since the circumference of a circle is 2 r =2 rad 2 r r 2p rad = = 360 One radian is an angle subtended by an arc of the same length as the radius! Monday, April 19, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 6
Unit of the Angular Displacement How many degrees are in one radian? 1 radian is 360 2prad 1rad =180 1 rad= o 57.3 p How radians is one degree? And one degrees is How many radians are in 10.5 revolutions? 3.141 180 2 o = = 1 1 0.0175rad 1 o 360 180 rad rev ( ) 10.5rev = 21 rad = 10.5 2 rev Very important: In solving angular problems, all units, degrees or revolutions, must be converted to radians! Monday, April 19, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 7
Example A particular bird s eyes can distinguish objects that subtend an angle no smaller than about 3x10-4 rad. (a) How many degrees is this? (b) How small an object can the bird just distinguish when flying at a height of 100m? (a) One radian is 360o/2 . Thus 4 3 10 rad = ( 360 2 rad (b) Since l=r and for small angle arc length is approximately the same as the chord length. r = 100 m 2 3 10 ( ) 3 10 rad ) = 4 o o 0.017 l = = 4 3 10 m = rad 3 cm Monday, April 19, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 8
Ex. Adjacent Synchronous Satellites Synchronous satellites are put into an orbit whose radius is 4.23 107m. If the angular separation of the two satellites is 2.00 degrees, find the arc length that separates them. What do we need to find out? The Arc length!!! =Arc length Radius =s (in radians) r = 2 rad 360deg Convert degrees to radians 0.0349 rad 2.00deg r =( 1.48 10 m (920 miles) )( ) s = 4.23 10 m 0.0349 rad 7 = 6 Monday, April 19, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 9
Ex. A Total Eclipse of the Sun The diameter of the Sun is about 400 times greater than that of the moon. By coincidence, the sun is also about 400 times farther from the earth than is the moon. For an observer on the earth, compare the angle subtended by the moon to the angle subtended by the sun and explain why this result leads to a total solar eclipse. (in radians) Arc length Radius = s r = I can even cover the entire sun with my thumb!! Why? Because the distance (r) from my eyes to my thumb is far shorter than that to the sun. Monday, April 19, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 10
Angular Displacement, Velocity, and Acceleration Using what we have learned earlier, how would you define the angular displacement? = f i f i = How about the average angular speed? Unit? rad/s Dimension (poll 5)? [T-1] f t t t f i i d lim = And the instantaneous angular speed? Unit? rad/s Dimension (poll 5)? [T-1] t dt 0 t By the same token, the average angular acceleration is defined as Unit? rad/s2 Dimension (poll 5)? [T-2] f i = t t t f i d And the instantaneous angular acceleration? lim = Unit? rad/s2 t dt 0 t When rotating about a fixed axis, every particle on a rigid object rotates through the same angle and has the same angular speed and angular acceleration. Monday, April 19, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 11
Rotational Kinematics The first type of motion we have learned in linear kinematics was under the constant acceleration. We will learn about the rotational motion under constant angular acceleration, because these are the simplest motions in both cases. Just like the case in linear motion, one can obtain Angular velocity under constant angular acceleration: Linear kinematics Angular displacement under constant angular acceleration: Linear kinematics One can also obtain Linear kinematics Monday, April 19, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 12
Problem Solving Strategy Visualize the problem by drawing a picture. Write down the values that are given for any of the five kinematic variables and convert them to SI units. Remember that the unit of the angle must be radians!! Verify that the information contains values for at least three of the five kinematic variables. Select the appropriate equation. When the motion is divided into segments, remember that the final angular velocity of one segment is the initial velocity for the next. Keep in mind that there may be two possible answers to a kinematics problem. Monday, April 19, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 13
Ex. Rotational Kinematics A wheel rotates with a constant angular acceleration of 3.50 rad/s2. If the angular speed of the wheel is 2.00 rad/s at ti=0, a) through what angle does the wheel rotate in 2.00s? Using the angular displacement formula in the previous slide, one gets 1 2 = + 2 t t f i 1 ( . 2 ) 2 = = + 11 0 . rad . 2 00 . 2 00 . 3 50 00 2 11 0 . = rev = . . 1 75 . rev 2 Monday, April 19, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 14
Example for Rotational Kinematics cntd What is the angular speed at t=2.00s? Using the angular speed and acceleration relationship = + t = + = . 2 00 . 3 50 . 2 00 . 9 00 rad / s f i Find the angle through which the wheel rotates between t=2.00s and t=3.00s. How many revolution is it? 1 = + 11.0rad 2 t t Using the angular kinematic formula f i 2 13.50 2.00 2 13.50 2 = t == + 2.00 2.00 At t=2.00s 2 ( ) 2 t == + 2.00 3.00 = 3.00 21.8rad At t=3.00s 3 10 8 . = 2 = 10 8 . rad Angular displacement = rev = . . 1 72 . rev 2 Monday, April 19, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 15
Relationship Between Angular and Linear Quantities What do we know about a rigid object that rotates about a fixed axis of rotation? Every particle (or masslet) in the object moves in a circle centered at the same axis of rotation. When a point rotates, it has both the linear and angular components in its motion. What is the linear component of the motion you see? Linear velocity along the tangential direction. How do we relate this linear component of the motion with angular component? r = So, the tangential speed vis The direction of follows the righthand rule. = r dl dt d d dt v ( ) l = = = The arc-length is r rdt What does this relationship tell you about the tangential speed of the points in the object and their angular speed?: Although every particle in the object has the same angular speed, its tangential speed differs and is proportional to its distance from the axis of rotation. The farther away the particle is from the center of Monday, April 19, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu rotation, the higher the tangential speed. 16
Is the lion faster than the horse? A rotating carousel has one child sitting on a horse near the outer edge and another on a lion halfway out from the center. (a) Which child has the greater linear speed? (b) Which child has the greater angular speed? (a) Linear speed is the distance traveled divided by the time interval. So, the child sitting at the outer edge travels more distance within the given time than the child sitting closer to the center. Thus, the horse is faster than the lion. (b) Angular speed is the angle traveled divided by the time interval. The angle both the children travel in the given time interval is the same. Thus, both the horse and the lion have the same angular Monday, April 19, 2021 speed. PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 17
How about the acceleration? How many different linear acceleration components do you see in a circular motion and what are they? Two Tangential, a at t, and the radial acceleration, a ar r. r v = dv dt Since the tangential speed v v is The magnitude of tangential acceleration atis d dt d dt = r ( ) ta = = = r r Although every particle in the object has the same angular acceleration, its tangential acceleration differs proportional to its distance from the axis of rotation. v2 = What does this relationship tell you? ( ) 2 r = r 2 r a The radial or centripetal acceleration aris = r r What does this tell you? The father away the particle is from the rotation axis, the more radial acceleration it receives. In other words, it receives more centripetal force. a r t a a + = ( r = ( ) ) 2 Total linear acceleration is 2 2 2 + r = r + 2 2 4 Monday, April 19, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 18
