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Trang 1Lecturer: Tran Thi Ngoc Dung
dungttn@gmail.com
HCMUT
Trang 2A B
0 v
v v
: is B to relative
A of velocity
The
v v
or v
v : sliding Without
bar
on the is
B and sphere
on the is
A where
same the
are B and
A points adjacent
two the
of s velocitie the
if
bar the
to relative sliding
not is
sphere The
ground /
B ground
/ A B
/
A
B A
ground /
B ground
/ A
What does it mean “Without sliding”?
Ground
0 v
and v
v
: sliding' '
is there If
B / A B
A
Trang 3Rotation about a moving axis
= Rolling
A
B
Ground
CM
cm
v
A cylinder is rolling on the ground
Two types:
- Rolling with sliding
- Rolling without sliding
Trang 4v
A
B
CM
of a point of the object
Rotation
about the axis through this point
of the center of mass
Rotation
about the axis through CM
of point A
Rotation
about the axis through A
Trang 5Velocity in rolling motion
CM v
CM through
axis about the
rotation
in the M
point a
of Velocity
v v
v
object.
rolling the
of M point a
of Velocity
cm
/
M
ground /
cm cm
/ M ground
/
M
cm
v
A
B
Ground
CM
cm / A
v
R
| v
D | v D/cm | R
cm / B
v
E
2
R
| v
cm / E
v
Trang 6v
A B
Ground
CM
cm / A
v
D
cm / D
v
ground /
cm cm
/ M ground
/
CM
v
gr / D
v
Rolling without sliding,
0
v A
because the ground is not moving, vB =0
0 v
R
) 1 ( 0
v v
v
cm
cm cm
/ A A
R
cm
cm
a R
v
R
CM
v
: angular velocity in rotational motion
: angular acceleration
Vcm ,acm : speed and acceleration of the CM
Projecting (1)on the x axis : x
y
Trang 7Rolling Translation
of the center of mass
Rotation
about the axis through CM
How to Solve Problems ‘Rolling without Sliding’
Method 1
ext , net
a
m
Step 1 Write Eq of Motion of the CM
Step 2 Write Eq of Rotation about the axis through CM
Step 3 Find relation between a cm and
cm / ext , net cm
Trang 8Translation
of point A having vA =0
Rotation
about the axis through A
How to Solve Problems ‘Rolling without Sliding’
Method 2
Step 1 Write Eq of Rotation about the axis through A =>
Step 2 Find relation between a cm and =>a cm
A / ext , net A
Trang 9A cylinder of mass m, radius R rolls without sliding on an inclined plane Find
a) acceleration of CM
b) angular acceleration
c) friction force
d) condition for having ‘rolling without sliding’
Method 1.
A
B
CM
W
N
f
x
y
Trang 10cm friction
2 cm
cm cm
2 cm
friction cm
2 cm cm
cm
friction cm
friction cm
friction cm
ma sin
mg f
: ) 2
(
from
R
I m
sin
mg a
sin mg a
) R
I m ( : ) 6
(
)
3
(
) 6 ( f
a R
I : (5) (4),
From
) 5 ( R
a
: and a
between lation
Re
.
3
Step
) 4 ( R
f I
: CM though
axis about the
Rotation of
Eq.
Step2.
) ' 3 ( cos mg
N )
3 ( N
cos mg
0
:
y
/
)
1
(
) 2 ( f
sin mg ma
:
x
/
)
1
(
axis
y and
on x ojecting
Pr
) 1 ( f
N g
m a
m :
CM of
motion of
Eq.
Step1.
A
B
CM
W
N
f
x y
Trang 11
sin
mg 2
1 f
; sin
g 2
1
a
mR I
: Hoop
b)
sin
mg 7
2 f
; sin
g 7
5
a
mR 5
2 I
: sphere solid
b)
sin
mg 3
1 f
; sin
g 3
2
a
mR 2
1 I
: cylinder
solid
a)
friction cm
2 cm
friction cm
2 cm
friction cm
2 cm
A
B
CM
W
N
f
x y
s s
s fric
max s fric
3 tan
cos mg sin
mg 3 1 Cylinder
N f
f f
' sliding without
rolling '
having for
condition
•
cm friction
cm
2 cm cm
ma sin
mg f
cos mg
N
R / a
R
I m
sin mg
a
Trang 12A cylinder of mass m, radius R rolls without sliding on an inclined plane
Find
a) acceleration of CM
b) angular acceleration
c) friction force
d) condition for having ‘rolling without sliding’
Method 2.
A
B
CM
W
N
f
x
y
sin
g 3
2 R
a
sin
g R 3 2
mR 2
3 mR
I I
: Theory Axis
Parallel
I
sin mgR sin
mgR
I
: CM though axis
about the Rotation
of
Eq.
cm
2 2
cm A
A
A
g sin 3
2
acm
R 3 2
R f
I ) b or
f sin
mg ma
) a
from f
Find
friction cm
friction cm
fric.
Trang 13Rolling without and with sliding
s N W
R a
R v
N f
: sliding with
rolling For
k fric
cm cm
k friction
zero is
force friction
static of
Work
R a
R v
max f
f f
: sliding without
rolling
For
cm
cm
s static
friction
Trang 14Kinetic Energy in Rolling
throughCM axis
about the
Motion Rotational
of E K
2 cm
CM of
motion nal
Translatio of
K.E
2
2
1 mv
2
1
0.40) (Answer slipping?
without
roll ill cylinder w the
for which friction
static
of
t coefficien the
of value minimum
the
is
What
50°.
=
at inclined plane
a down rolls
cylinder
A
Homework
) 2µ
= tan (b) , mgsin 2
1 f (a) (Answers
slipping?
without roll
will hoop
the for which tan
of value maximum
the is what (b)
friction, of
force the
is what (a)
incline,
an down rolling
hoop a
For 2)
s