Transcript 11 rotation

11 rotation
The rotational variables
Relating the linear and angular variables
Kinetic energy of rotation and rotational inertia
Torque, Newton’s second law for rotation
Work and rotational kinetic energy
11-1,--11-4 the rotational variables:
(1) Translation and rotation:
Rigid body（刚体）: is a body (that can rotate) with
all its parts locked together and without any change in
its shape.
In pure translation（平动）, every point of the
body moves in a straight line , and every point
moves through the same linear distance during a
particular time interval.
In rotation about the fixed axis（定轴转动）,
every point moves through the same angle during a
particular time interval.
(2) The rotation of a rigid body about a fixed axis:
Angular position: to describe the rotation of a rigid
body about a fixed axis, called the rotation axis,we
assume a reference line is fixed in the body,
perpendicular to that axis and rotating with the body.
We measure the angular position  of this line
relative to a fixed direction (x).
s
s

r
Unit of : radian( rad) 弧度
Always: =(t)


o
r
s
x
Angular displacement: a body that rotates about a
rotation axis, changing its angular position from 1
to 2 , undergoes an angular displacement:
   2 1
Where  is positive for counterclockwise rotation
and negative for clockwise rotation.
Angular velocity and speed:
Average angular velocity:
(Instantaneous) angular velocity:
 avg


t
d

dt
avg and  are vectors, with
directions given by the right-hand
rule. They are positive for counterclockwise rotation and negative for
clockwise rotation.
The magnitude of the body’s angular
velocity is the angular speed.


Angular acceleration:
Average angular acceleration:
 avg


t
d
(Instantaneous) angular acceleration:  
dt
Both  avg and  are vectors.
When  is positive:
>0, the direction is the same as ;
<0, the direction is reversed to .


(3) Rotation with constant angular acceleration
  const.
   0   t

2
1
(   0 )   t  2  t
 2
   02  2 (   0 )
See page 221: Table 11-1 equations of motion for
constant linear acceleration and for constant angular
acceleration
11-5 relating the linear and angular variables
(1) Circular motion: description by linear variables
Distance: S=r or s=r
v
ΔS
R Δθ
0
θ
Speed:
ω,
x
s ds
v  lim

t  0  t
dt

v (t )

v ( t  t )
B
R

O

v (t  t )
A
X

v

v (t )
v n

v t
内法向 n

t

 v
a

t
v t ˆ v n
t
nˆ
t
t
 a t tˆ  a n nˆ
切向
vn
v

AB r

v

v (t  t )

v (t )
v AB
vn 
r
v n

v t
vn
v AB v
an  lim
 lim

t 0 t
r
t 0 r t
2
Radial acceleration:
v t  v(t  t )  v(t )
Tangential acceleration:
vt dv
at  lim

t  0 t
dt
(2) Relating the linear and angular variables
s  r
ds d
v

r  r
dt dt
dv d
at 

r  r
dt
dt
Home work: 5E, 29p
2
v
2
an    r
r
11-6, 11-7 kinetic energy of rotation and rotational inertia
(1)rotational kinetic energy:
1 2
K  I
2
(Compare with kinetic
energy of particle)
Proof: treat the rotation rigid body as a collection of
particles with different speeds. We can then add up
the kinetic energies of all the particles to find the
kinetic energy of the body as a whole. In this way
we obtain the kinetic energy of a rotating body,
1
1
1
2
2
2
2
K   mi vi   mi (ri )  ( mi ri )
2
2
2
z
Let:
I   mi ri
vi
2
ri
O
Then:

mi
1 2
K  I
2
I is rotational inertia (or moment of inertia)(转动惯量）
(2) Rotational inertia :
I   mi ri
2
For rigid body with continuously distributed mass:
I   r dm
2
Where r and ri represent
the perpendicular distance
from the axis of rotation
to each mass element in
the body.
For particle:
I=mr2
dm
M
r
See page 227 table 11-2 some rotational inertias
(calculate (e):
1. thin rod about axis through center perpendicular to
length;
2. About axis through the end of the rod perpendicular
to length.)
Sample problem:11-5
Unit: kg.m2
(3) Parallel-Axis Theorem:
I  I com  Mh
2
Where I is the rotational inertia of a
body of mass M about a given axis.
I com is the rotational inertia of the body
about a parallel axis that extends
through the body’s center of mass.
h is the perpendicular distance between
the given axis and the axis through the
center of mass.
M is the mass of rigid body.
ICom
C
I
h
平行
M
Proof of the parallel-axis theorem: (by yourself)
Question: What does the rotational inertia relate to?
Answer: Mass and its distribution.
11-8 torque
Torque is a turning or twisting
action on a body about a rotation
axis due to a force.
z
F

  rF

r
  rF sin   rFt
O ×
Where Ft is the component of F
perpendicular to r , and  is
the angle between
and
.
r
F
Unit: N.m
P
Caution:
(1) Torque is a vector for rotation about fixed axis,
its direction always along the axis, either positive
or negative. (see page 230)
(2) If several torque act on a rigid body that rotate
about a fixed axis, the net torque is the sum of
individual torque.
(3) The net torque of internal forces is zero.
11-9 Newton’s second law for rotation (转动定律）
Compare with:
We obtain:
F  ma
 net  I 
For rotation about fixed axis:
Proof of equation:
 net  I
z
ω,α
v
ri
O
Treat the rigid body as a
collection of particles, F i
and f i are the external
and internal forces of
mass element mi ,thus:
Fi

Δmi

fi
Then:
F i  f i  mi a
Fit  f it  mi at
Fit ri  f it ri  mi at ri  mi ri 
2
For whole rigid body:

Fit ri   fit ri   mi ri 
2

That is:
i
 0  I
 net  I
 net  I
Sample problem: 11-7, 11-8
11-10 Work and rotational kinetic energy
(1) work:
From:
dW  F  d r
For rigid body:
dW  Ft ds  Ft  r  d    d
From initial angular position to final angular position,
the work is:

W
   d
0
Caution:
(1) If torque is constant, then:
W   
(2) When several torques act on the rigid body,
the net work is sum of individual work.
(3) This work is scalar also.
(2) Work-kinetic energy theorem:
d
W     d   I  d    I
d
dt
f
1 2 1 2
  I  d  I f  I i
i
2
2
 K f  K i  k
(3) power:
dW d
P

 
dt
dt
(4) The gravitational potential energy of rigid body:
U  Mghc
(1) U can not change when the rigid body rotate about
axis that extends through the body’s center of mass.
(2) The conservation of Mechanical energy and
conservation of energy can also be used for rigid body.
Sample problem: 11-9, 11-10
Home work: 29p, 65p, 66p