Chapter 10 & 11
Rotational Motion About
a Fixed Axis
Torque,Rotational Inertia and Law
Work and Rotational Kinetic Energy
Angular Momentum and
Conservation of Angular Momentum
1,Angular Quantities (P235-238)
1.Angular position?,
2.Angular Displacement(角位移 ):
Where is positive for counterclockwise rotation
and negative for clockwise rotation.
Always,? =?(t)
12
ttt d
dlim
0
3.Angular Velocity,
Right-hand-rule,When the fingers of the right hand
are curled around the rotation axis and point in the
direction of the rotation,then the thumb points in the
direction of,(p241)?
ttt d
dl i m
0
4.Angular Acceleration,
rdtddtds
rv
or
2,Relation of Linear and Angular Variables
The distance along a circular arc:
The linear speed (with r held constant):
rs m e a s u r e ),( r a d i a n
The tangential component at,
rt
The radial component an:
2
2
rv
r
va
n
-?0? =?0 +?t
-?0 =?0t +?t2
t?2 =?02 +2?(? -?0)
-?0 = (?0 +?)t
0? -?0 =?t -?t2
v = v0 + at x -x0
x - x0 = v0t + at2 v
v2 = v02 + 2a(x - x0) t
x - x0 = (v0 + v)t a
x - x0 = vt - at2 v0
Equations of Motion for Constant Linear Acceleration are
analogous to the ones of Constant Angular Acceleration:
Linear Missing Angular
Equation variable Equation
2
1
2
1
2
1
2
1
2
1
2
1
3,Kinematics Equations for Uniformly
Accelerated Rotational Motion(p238)
4 Purely Rotational motion
Any point moves around one same axis.
Axis of rotation
FR
5,Torque on an axis
s i nRF?
FR
The torque about a given axis is defined as:
RF
6,N-II Law for Rotation (定轴转动定律 )
In e t? N-II Law for Rotation
—— Net torque on a rigid body equals to the
product of rotational inertia and angular
acceleration it causes about same axis.
the moment of inertia or rotational
inertia (转动惯量 ) on the axis. 2ii rmI
The moment of inertia I is a measure of the
rotational inertia of a body,which plays the
same role for rotational motion that mass does
for translational motion.
dl
dS
dV
dm
where,. dmrI 2
7,The Calculation of Rotational Inertia of Body
质量连续分布刚体转动惯量的计算方法：
①,确定刚体的质量密度。
②,建立坐标系，坐标原点为轴。
③,确定质量元 dm。
④,由定义计算。
8 Solving Problems in Rotational Dynamics
In e t? N-II Law for Rotation
1,As always,draw a clear and complete diagram.
2,Draw a free-body diagram.
3,Identify the axis of rotation and calculate the
torques about it.
4,Apply Newton’s second law for rotation and for
translation.
5,Solve the resulting equations.
.21 2 IK
9,Rotational Kinetic Energy (P254-255)
21 dMW
力矩的功
KIIW if 22 2121 f
i
dWwhere
work-kinetic energy theorem for rigid body
rotated about a fixed axis:
—– the work done on a rigid body during a
finite angular displacement equals to the
change of its rotational kinetic energy.
KWKW
PFvP
IKmvK
dWF dxW
Im a F
Im
dtddtdva
dtddtdxv
x
n etn et
t h eorem en ergy
k n i et i c-w ork
f orce)-p ow er( c
en ergy k i n et i c
w ork
l aw 2n d
m ass
/ / onaccel erat i
/ / vel oci t y
p osi t i on
rot at i on p u re nt ran s l at i o p u re
22
2
1
2
1
Some corresponding relations between translation and
rotational motion:
The torque acting on a particle relative to a fixed
point O is a vector quantity defined as:
Fr
10,Torque about a point and particles (P277)
torque? on a system of particles will be:
ii Fr
vmP
The linear momentum of a particle:
11,Angular momentum of a Particle(p277)
The Angular momentum relative to a fixed point：
)( vrmPrl
Magnitude：
s i nm v rll
Direction,right-hand rule
In linear motion,we have Newton’s
second law for a single particle:,dt
pdF
n et
one can guess for its angular form,
dt
ld
n e t
The time rate of change of angular momentum of
a particle is equal to the net torque applied to it.
12,Relation between Angular Momentum
and Torque:(p278)
The rotational equivalent of N-II law for a particle.
If net external torque acting on particle’s system
is zero,the total angular momentum of the system
remains conserved,
,0If CLn et
13,The Conservation Law of Angular Momentum (p284)
14,Angular Momentum & Torque of Rigid Body
(P280-281)
i
iiii
i
i rmrmL )(
2v
O ir?
im
iv
IL?
z
Summing over all the
particles to obtain:
dtLdn e t
Using the angular momentum principle of
particle’s system,
For a rigid body,the component along the
rotation axis is:
dtdLa x i s
I
dt
dII
dt
d )(
刚体所受的外力矩等于刚体角动量的变化率。
2、刚体定轴转动的角动量定理
16,Conservation of Angular Momentum (p251-252)
c o n st a n tL fi LL?
or,ffii II
If no net external torque acts on a rigid body,
IdtdLa x i s
The total angular momentum of a rotating
body remains constant if the net external torque
acting on it is zero.
Rotational Motion About
a Fixed Axis
Torque,Rotational Inertia and Law
Work and Rotational Kinetic Energy
Angular Momentum and
Conservation of Angular Momentum
1,Angular Quantities (P235-238)
1.Angular position?,
2.Angular Displacement(角位移 ):
Where is positive for counterclockwise rotation
and negative for clockwise rotation.
Always,? =?(t)
12
ttt d
dlim
0
3.Angular Velocity,
Right-hand-rule,When the fingers of the right hand
are curled around the rotation axis and point in the
direction of the rotation,then the thumb points in the
direction of,(p241)?
ttt d
dl i m
0
4.Angular Acceleration,
rdtddtds
rv
or
2,Relation of Linear and Angular Variables
The distance along a circular arc:
The linear speed (with r held constant):
rs m e a s u r e ),( r a d i a n
The tangential component at,
rt
The radial component an:
2
2
rv
r
va
n
-?0? =?0 +?t
-?0 =?0t +?t2
t?2 =?02 +2?(? -?0)
-?0 = (?0 +?)t
0? -?0 =?t -?t2
v = v0 + at x -x0
x - x0 = v0t + at2 v
v2 = v02 + 2a(x - x0) t
x - x0 = (v0 + v)t a
x - x0 = vt - at2 v0
Equations of Motion for Constant Linear Acceleration are
analogous to the ones of Constant Angular Acceleration:
Linear Missing Angular
Equation variable Equation
2
1
2
1
2
1
2
1
2
1
2
1
3,Kinematics Equations for Uniformly
Accelerated Rotational Motion(p238)
4 Purely Rotational motion
Any point moves around one same axis.
Axis of rotation
FR
5,Torque on an axis
s i nRF?
FR
The torque about a given axis is defined as:
RF
6,N-II Law for Rotation (定轴转动定律 )
In e t? N-II Law for Rotation
—— Net torque on a rigid body equals to the
product of rotational inertia and angular
acceleration it causes about same axis.
the moment of inertia or rotational
inertia (转动惯量 ) on the axis. 2ii rmI
The moment of inertia I is a measure of the
rotational inertia of a body,which plays the
same role for rotational motion that mass does
for translational motion.
dl
dS
dV
dm
where,. dmrI 2
7,The Calculation of Rotational Inertia of Body
质量连续分布刚体转动惯量的计算方法：
①,确定刚体的质量密度。
②,建立坐标系，坐标原点为轴。
③,确定质量元 dm。
④,由定义计算。
8 Solving Problems in Rotational Dynamics
In e t? N-II Law for Rotation
1,As always,draw a clear and complete diagram.
2,Draw a free-body diagram.
3,Identify the axis of rotation and calculate the
torques about it.
4,Apply Newton’s second law for rotation and for
translation.
5,Solve the resulting equations.
.21 2 IK
9,Rotational Kinetic Energy (P254-255)
21 dMW
力矩的功
KIIW if 22 2121 f
i
dWwhere
work-kinetic energy theorem for rigid body
rotated about a fixed axis:
—– the work done on a rigid body during a
finite angular displacement equals to the
change of its rotational kinetic energy.
KWKW
PFvP
IKmvK
dWF dxW
Im a F
Im
dtddtdva
dtddtdxv
x
n etn et
t h eorem en ergy
k n i et i c-w ork
f orce)-p ow er( c
en ergy k i n et i c
w ork
l aw 2n d
m ass
/ / onaccel erat i
/ / vel oci t y
p osi t i on
rot at i on p u re nt ran s l at i o p u re
22
2
1
2
1
Some corresponding relations between translation and
rotational motion:
The torque acting on a particle relative to a fixed
point O is a vector quantity defined as:
Fr
10,Torque about a point and particles (P277)
torque? on a system of particles will be:
ii Fr
vmP
The linear momentum of a particle:
11,Angular momentum of a Particle(p277)
The Angular momentum relative to a fixed point：
)( vrmPrl
Magnitude：
s i nm v rll
Direction,right-hand rule
In linear motion,we have Newton’s
second law for a single particle:,dt
n et
one can guess for its angular form,
dt
ld
n e t
The time rate of change of angular momentum of
a particle is equal to the net torque applied to it.
12,Relation between Angular Momentum
and Torque:(p278)
The rotational equivalent of N-II law for a particle.
If net external torque acting on particle’s system
is zero,the total angular momentum of the system
remains conserved,
,0If CLn et
13,The Conservation Law of Angular Momentum (p284)
14,Angular Momentum & Torque of Rigid Body
(P280-281)
i
iiii
i
i rmrmL )(
2v
O ir?
im
iv
IL?
z
Summing over all the
particles to obtain:
dtLdn e t
Using the angular momentum principle of
particle’s system,
For a rigid body,the component along the
rotation axis is:
dtdLa x i s
I
dt
dII
dt
d )(
刚体所受的外力矩等于刚体角动量的变化率。
2、刚体定轴转动的角动量定理
16,Conservation of Angular Momentum (p251-252)
c o n st a n tL fi LL?
or,ffii II
If no net external torque acts on a rigid body,
IdtdLa x i s
The total angular momentum of a rotating
body remains constant if the net external torque
acting on it is zero.
