chapter 6 rotation motion
TRANSCRIPT
Chapter 6 Rotation Motion
Angular Position, Velocity, and Acceleration
Rotational Kinetic Energy
Torque and Rotational Dynamics
Rotational Plus Translational Motion ; Rolling
Angular Momentum of a Particle
Conservation of Angular Momentum
Angular Momentum of a Rotating Rigid Object
2 1
2 1 t t t
6-1 Angular Position, Velocity, and Acceleration*
0lim
t
d
t dt
2 1
2 1t t t
0lim
t
d
t dt
10-04
rv
P
/S r 0( ) /180 (deg)rad
x
y
Unit: : /rad s : /f rev s; 2 f
P. 250 CCU Physics 6 - 1
d d
v R Rdt dt
t
dv da R R
dt dt
22
r
va R
R
2 4R
ta
P
x
y
ra
a
2 2
t ra a a
CCU Physics 6 - 2
2
0 0
1
2x x v t at
2 2
0 0 2 ( - )v v a x x
0v v at 0 t
2
0 0
1
2t t
2 2
0 02 ( )
Constant Angular
AccelerationConstant Acceleration
;d d
dt dt
;
dx dvv a
dt dt
CCU Physics 6 - 3
Moment of inertial
(轉動慣量 )
2
i i
i
I m r
21
2RK I
6-2 Rotational Kinetic Energy ( Rotation about a Fixed Axis)
21
2R i iK m v
Rotational
Kinetic Energy
2 21
2i im r ω
CCU Physics 6 - 4
The moment of inertial of a body is a measure of its
rotational inertia, that is ,its resistance to change in its
angular velocity.
If MA= MB= MC
IC > IB > IA
IA > IB
AB
dmrI 2
For continuous Bodies
CCU Physics 6 - 5
Example 6-1 Find the moment of inertia of a uniform thin hoop of
mass M and radius R about an axis perpendicular to the plane of the
hoop and passing through its center.
2 2R dm MR
2
zI r dm
P. 184 CCU Physics 6 - 6
Example 6-2 Calculate the moment of inertia of a uniform rigid rod
of length L and mass M about an axis perpendicular to the rod (the
y axis) and passing through its center of mass.
21
12ML
Mdm dx dx
L
/ 22 2
/ 2
L
yL
MI r dm x dx
L
/ 22
/ 2
L
L
Mx dx
L
CCU Physics 6 - 7
Example 6-3 (a) Show that the moment of inertia of a uniform
hollow cylinder of inner radius R1, outer radius R2, and mass M, is
½ M(R12 + R2
2), if the rotation axis is through the center along the
axis of symmetry. (b) Find the moment of inertia for a solid cylinder.
2 .dm dV hR dR
2
1
2 32R
RI R dm hR dR
4 44 42 12 12 ( ).
4 2
R R hh R R
2 2 2 2 2 2
2 1 2 1 1 2
1( )( ) ( ).
2 2
hI R R R R M R R
2 0R R
2 2
2 1( ) .M V R R h
1 0R For a solid cylinder, and 2
0
1.
2I MR
CCU Physics 6 - 8
2
0
1
2CMI MR
2
0CMI MR
21
12CMI M
6-3 Parallel-axis theorem (平行軸定理) : I = ICM + MD2
2
02I MR
Axis
Axis
Axis
2
0
3
2I MR
21
3I M
CCU Physics 6 - 9
m 2m 4m
0 1 2 3 4
O O’’ O’1 1 2 2 4 4
31 2 4
i i
cm
i
m xx
m
1 ( 2) 2 ( 1) 4 10
1 2 4
i i
cm
i
m xx
m
1 0 2 1 4 32
1 2 4
i i
cm
i
m xx
m
Position of the center of mass
from difference origin
Position of the center of mass from the
center of mass is always zero. 0 cmx x dm
在質心座標系統的質心位置為零。
CCU Physics 6 - 10
Proof of Parallel-Axis Theorem (平行軸定理)
2 2( )I x y dm ' 2 ' 2 ( ) ( ) x yD x D y dm
2 2 2 2 ( ) ( ) x yD D dm x y dm 2 2x yD x dm D y dm
2 CMI MD I
0 0
0 x dm 在質心座標系統的質心位置為零
CCU Physics 6 - 11
Example 6-4 A uniform disk (radius R) rotates freely
about a horizontal axis P. What is the angular velocity of
the disk as the disk passes through the vertical position.
P
m
?
2 2 21 3
2 2PI MR MR MR
2 2 21 3
2 4PMgR I MR
4
3
g
R
CCU Physics 6 - 12
Example 6-5 A T-shape stick rotates freely about a
horizontal axis P. What is the angular velocity of the stick
as the stick passes through the vertical position?
( : 600 00 )
2 2 2 21 1 17
3 12 12PI ML ML ML ML
0 0 211 cos60 1 cos60
2 2P
LMg MgL I
36
34
g
L
2 23 17
4 24MgL ML
L
P
?
mm
L
1 cosh L
h
CCU Physics 6 - 13
6-4 Perpendicular-Axis Theorem : Iz = Ix + Iy
2 2( )zI x y dm 2 2 x dm y dm
y xI I 2 2 2 2 zI x z dm y z dm
( z = 0 for flat objects )
Valid only for flat objects
The sum of the moments of inertia of a plane object about any two
perpendicular axes in the plane of the object , is equal to the
moment of inertia about an axis through their point of intersection
perpendicular to the plane of the object.
CCU Physics 6 - 14
MR ?xI
21
2zI MR
2 21 1 1 1
2 2 2 4x zI I MR MR
x yI I (symmetry)
2z x y xI I I I
CCU Physics 6 - 15
6-5 Rotational Plus Translational Motion ; Rolling**
CM
ds dv R R
dt dt
CMCM
dv da R R
dt dt
Condition for pure rolling motion
S R
P. 267
CMv R CMa R
Pure rolling motion
;
CCU Physics 6 - 16
一轉動中物理之角速度，相對於其上各點均相同。
CCU Physics 6 - 17
The motion of a rolling object can be modeled
as a combination of pure translation and pure
rotation.
translation rotation rolling
CCU Physics 6 - 18
For pure rolling all points move in a direction
perpendicular to an axis through the instantaneous
point of contact P. All points rotate about P.
v r
r
CCU Physics 6 - 19
A object that both translational and rotational motion also has both
translational and rotational kinetic energy:
2 21 1
2 2tot CM CMK Mv I
A ring, a disk and a solid
sphere have the same
radius and same mass
Two solid sphere with
different radius
Which arrives at the bottom first ?
CMv
6-6 Rotational Kinetic Energy ( General Case )
CCU Physics 6 - 20
2 21 1' 0;
2 2i i i i C
i i
m v m v I
2 21 1
2 2CM C
K Mv I
21
2i iK m v
21( )
2i CM im v v
21 1 12
2 2 2i CM i i i CM i
i i
K m v m v m v v
21 1( )
2 2CM i i CM i i
i i
K Mv m v v m v
'
i cm ir r r
i CM iv v v ;
Proof
CCU Physics 6 - 21
21
2pK I
Method 2 :
2 21 1
2 2CM CMK Mv I
Method 1 :
Find the kinetic energy of a rolling disk
Vc
ω
P
2 21( )
2CMI MR
2 21 1
2 2CM CMI Mv
CCU Physics 6 - 22
2 21 1
2 2CM CMK Mv I
Example 6-6 What will be the speed of a disk of mass M and
radius R when it reaches the bottom of an incline if it starts from
rest at a vertical height h and rolls without slipping ?
23
4CMK Mv Mgh
4 v
3CM
gh
2 2 21 1 1( )
2 2 2
CMCM
vMv MR
R
h
CCU Physics 6 - 23
6-7 Torque (力矩 )
t : Torque
d : level arm
sinR F RFt
Torque is a vector, but we will consider only its magnitude
here and explore its vector nature in chapter 11.
1 2 1 1 2 2net F R F Rt t t t The sign of the torque is positive if the
turning tendency is counterclockwise
10-14
1F
2F
1R
2R
CCU Physics 6 - 24
2
i i i
i i
m Rt I
inti ext extt t t t
2
i i im Rt
ext It 2
i iiI m R
6-8 Rotational Dynamics
/ /
i i it i iF m a m R
iR im
iF / / 2
i i i iR F m R
CCU Physics 6 - 25
ext
CM CMIt
Even center of mass reference frame is noninertial
m 2m1x
2x
cm
F
1fF
2fF
m 2m1x
2x
F
1fF
2fF
( )a ( )b
a aInertial Force ( Ff )
1
22
f
f
F ma
F ma
2 1 1 2 2 2( )CM Fx m gx m gx Fxt
2 1 1 2 2 2( )Fx m gx m gx Fxt
Case (a)
Case (b)
The net torque, relative to the center of mass, contributed by
the inertial force is always equal to zero.
0;a g g a
CCU Physics 6 - 26
CMCM
dL
dtt
Even center of mass reference frame is noninertial
The spin motion of the earth
CCU Physics 6 - 27
RM
1m 2m
1T 2T
2 1m m
Example 6-7 Find the acceleration of m2 (ICM = ½ MR2)
a) Draw free body diagram for
the pulley and each block .
b) Write down the equations for
each diagram you draw in part
a). (Newton’sd 2nd law for
linear motion and rotational
motion)
c) What is the relation between a
and ?
d) Solve the acceleration a.
CCU Physics 6 - 28
RM
1T2T
2 2 2m g T m a
1 1 1T m g m a
2 1 CMT T R I
2 1 2
1
2
CMIT T a Ma
R
2 1
1 2
( )
/ 2
m m ga
m m M
1m 2m
1T 2T
1m g2m g
/a R
2 1m m
aa
CCU Physics 6 - 29
sin Smg f ma
Example 6-8 A sphere rolls without slipping down an incline.
(a) Find the linear acceleration of the CM. (b) what is the minimum
coefficient of friction required for the sphere to roll without slipping?
R CMf R I 2 2( )5
aMR
R
2
5Sf ma
5 sin
7a g
2 sin
7Sf mg
max
2 cos sin
7s sf N mg mg
2tan
7S
10-30
Sf
mg
N
x
y
= 0 fS = 0, a = 0
CCU Physics 6 - 30
SF f Ma
21
2S CM
aFr f R I MR
R F
rR
Sf
P2( )
3
R r Fa
R M
2
3S
R rf F
R
Example 6-9 A constant force (F) is applied to a yo-yo
(Icm = ½ MR2). The yo-yo rolls without slipping. What is the
acceleration of the yo-yo ? What is the minimum coefficient of
friction required for the yo-yo to roll without slipping
1
2S
rF f Ma
R
0;S
Ff a
M / 2r R
/ 2r R 0Sf
maxS Sf mg2
3S
R r F
R mg
CCU Physics 6 - 31
Fr
Rf
21
2S CM
af R Fr I MR
R
2( )
3
R r Fa
R M
2
3S
R rf F
R
SF f Ma
1
2S
rf F Ma
R
maxS Sf mg2
3S
R r F
R mg
CCU Physics 6 - 32
Example 6-10 A bowling ball of mass M and radius r0 is thrown
along a level surface so the initially it slides with a linear speed v0
but does not rotate. As it slides, it begin to spin, and eventually
rolls without slipping. How long does it take to begin rolling
without slipping ?
kmg ma
0k CMmg r I
ka g
0
2
5
kg
r
0CM kv v gt
0
2
5
kgt
r
0CMv r 02
7 k
vt
g
CCU Physics 6 - 33
What is the direction of the static friction force on the disk in the
figure (rolls without slipping)?
F
rR
21; 0.6
2CI mR r R
Direction of the static friction force (I)
Sf
C
C C
C C
C C
Fr FrRFr I R
I I
FF Ma a
M
if R a R v
(Spin too fast)
(points to the right )
Sf
21.2 1.2
Fr FR a a
MR M
Sf
Without static friction force
R Cv
CCU Physics 6 - 34
What is the direction of the static friction force on the disk
in the figure (rolls without slipping)?
21; 0.2 ; 2
2CI mR r R F mg
F
rR P
037
Direction of the static friction force (II)
20.8
FrR mg
mR
R a Spin too slowSf
cos sinF mg ma
4 32
5 5a mg mg mg
21
2cFr I mR
Sf
CCU Physics 6 - 35
6-9 The vector product and Torque
ˆ sin C A B AB n
A B B A
n̂
sinr F rFt
r
CCU Physics 6 - 36
( ) A B C A B A C
ˆˆ ˆ
i j k
A Bx y z
x y z
A A A
B B B
x y z z y
y z x x y
z x y y x
C A B A B
C A B A B
C A B A B
or
ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ; ;i j k j k i k i j
ˆ ˆˆ ˆ ˆ ˆ 0i i j j k k
C A B ˆ ˆˆ ˆ ˆ ˆx y z x y zA i A j A k B i B j B k
CCU Physics 6 - 37
L r p
dv dpF m
dt dt
( )dp d dL
r F r r pdt dt dt
t
Linear momentum p mv
Angular Momentum
0
( )d dr dp dp
r p mv r mv v rdt dt dt dt
note
11-04
6-10 Angular Momentum of a Particle
CCU Physics 6 - 39
( ) ( )'( ) ( )e e
i i ij i i
i i j i
dL
dtt t t t
12 21 1 12 2 21 1 2 12
1 2 12
( ) 0
[ ( ) // ]
r F r F r r F
if r r F
t t
r1
r2
r1 - r2
( )e ttot
dL
dtt
t iiL L0τ
(e)tot
if conserved
6-11 Conservation of Angular Momentum
( )e
i i ji i
dL
dtt t t
( ) ( )e e
tot i
i
t twhere
L conserv.
CCU Physics 6 - 40
The angular momentum of the particle relative to point O
1 1 1 1ˆL m v a z
O
1v
1m
1a2m
2v
3a
3v3m
x
y
2 0L
3 3 3 3ˆL m v a z
ˆL mvrzoz
CCU Physics 6 - 41
cosi i i i
i i
L L m rv
6-12 Angular Momentum of a Rotating Rigid Object
i i i iL m r v
L I
dL dI I
dt dt
ext It text
dL
dtt 11-03
2
i i i i i
i i
m R v m R
CCU Physics 6 - 42
p mv L I
P : Momentum L : Angular Momentum
dPF ma
dt
dLI
dtt
0 constantF P 0 constantLt Conservation of
Linear MomentumConservation of
Angular Momentum
CCU Physics 6 - 43
; / 2i fr R r R
Example 6-11 A circular platform of mass M and radius R rotates
about a frictionless vertical axle. A man of mass m walks slowly
from R to R/2. If the angular speed of the system is when the
man is at R, what is the angular speed when he reaches R/2 ? What
is the change of the kinetic energy of the system K? ( M = 4m )
i
22 2 21 1
2 2 4i f
mRMR mR MR
No external torque ( r = 0 ) Li = Lf
2 4 4
2 3f i i
M m
M m
2
2 2 2 2 21 9 4 1 13
2 4 3 2 2i i iK mR mR mR
Work done
by the man
CCU Physics 6 - 44
CMCM
dL
dtt
Even center of mass reference frame is noninertial
The spin motion of the earth
CCU Physics 6 - 45
CM i i iL m r v
CM i
i i i i i CMi i
dL dr dm v m r v v
dt dt dt
i CM ir r r
i CM iv v v ;
i
i i i i CMi i
dvr m m r v
dt
i ii
r F
CMCM
dL
dtt
x y
z
ri
CMr
ir
Proof (Option)
CCU Physics 6 - 46
Spin and Orbital Angular Momentum
( )i i CM i i i
m r v m r v i CM CM CM i i
i
m r v r m v 0
0
CM CM i iL r p r p 0 CM
L L L or
where CM CM i
i
p Mv M m Let
i i iL m r v
( ) ( )i CM i CM i
i
m r r v v xy
z
ri
CMr
iri CM i
r r r i CM i
v v v ;
CCU Physics 6 - 47
L I2( )
CML I Mh
2 CM
Mh I
0 CML L
2
CMI I Mh
Example 6-12 Find the angular momentum of the body
L0 : The angular moment of the CM motion about origin O
in an inertial frame (Orbital angular momentum)
LCM : The angular momentum relative to CM
(Spin angular momentun)
0 CML L L
CCU Physics 6 - 48
Example 6-13 Find the angular momentum relative to O of a pure
rolling disk (radius R) .
O vm
1 3ˆ ˆ
2 2L mvR mvR z mvRz
0 CML L L
ˆCM CMmr v I z
x
y
r
21ˆ ˆ
2
vmvRz mR z
R
CCU Physics 6 - 49
Example 6-14 A wheel is given an initial angular speed 0. A
kinetic friction force acts the on wheel as it initially skids across the
floor. Find the magnitude of the velocity of the wheel when the
wheel is rolling without slipping ( pure rolling ).
f
f CM
vmRv I
R
0
2
CM
f
CM
I Rv
I mR
, ,S i S fL L
0CM f CM fI mRv I
Choose O as the origin, the
torque by fk is equal to zero.
f
fv
S d
,CM Sr
R
0v
0
00v
Pure rolling
direction of L
CCU Physics 6 - 50
Example 6-15 : On a level billiards table a cue ball, initially at rest
at points O on the table, is struck so that it leaves the cue stick with
a center of mass speed v0 and a “ reverse” spin of angular speed 0.
A kinetic friction force acts the on ball as it initially skids across the
table. If 0 = 3 v0/R , determine the ball’s CM velocity when it
starts to roll without slipping ?
0
1
7V V
2 2
0 0
2 2( ) ( )5 5
MV R MR MVR MR
Choose O as the origin, t = 0 Li = Lf
v0
O
0
xy
2 20
0
32 2( ) ( )5 5
V VMV R MR MVR MR
R R
0
1 7
5 5MV R MVR
Direction of L
CCU Physics 6 - 51
What is the direction of the kinetic friction force on the disk in the
figure (rolls with slipping) ?
00 1.2
v
R
Direction of the kinetic friction force
0 0R v Spin too fast
(points to the right )
kf ma
kf
x
y
kf
Move faster
Spin slowerk CMf R I
v0
0
0R 0vP
Velocity of point P
relative to the ground
0Pv v R
0Pv kf
0Pv kf
CCU Physics 6 - 52
vm
1 4M m6R
Friction force
is negligible
R2 5M m
4R
Example 6-16 Describe the motion after the collision
( friction force is negligible )
No external force
Conservation of linear
momentum
Conservation of
angular momentum
about point O.
vcm = constant
Find the position of the
center of mass of the
system after collision ycm
Choose an origin (e.g. center of the ring )
tot CM cm CMmvr M v r I
ringO
CCU Physics 6 - 53
4 4 42
10CM
m R m Ry R
m
0.110
CM
mvv v
m
4iL mvR
After collision
2 2 2 2 2 21(2 ) (4 )(6 ) 4 (2 ) 5 ( ) 5 (2 ) 57
12CMI m R m R m R m R m R mR
2(10 )(2 ) 2 57f CM CM iL m R v I mRv mR L
2
2 2
57 57
mvR v
mR R ; 0.1CMv v( clockwise )
vm
1 4M m6R
R2 5M m
direction of L
CCU Physics 6 - 54
4 4 42
10CM
m R m Ry R
m
0.110
CM
mvv v
m
2iL mvR
After collision
2 2 2 2 2 21(2 ) (4 )(6 ) 4 (2 ) 5 ( ) 5 (2 ) 57
12CMI m R m R m R m R m R mR
257f CM iL I mR L
2
2 2
57 57
mvR v
mR R ; 0.1CMv v( clockwise )
vm
1 4M m6R
R2 5M m
P
Method 2 :
Angular moment relative to P
angular moment relative to P
( P : center of mass of the system ) direction of L
CCU Physics 6 - 55