11 rotation The rotational variables Relating the linear and angular variables Kinetic energy of rotation and rotational inertia Torque, Newton’s second.
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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:
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.
Rigid body(刚体) : is a body (that can rotate) with all its parts locked together and without any change in its shape.
(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).
r
s
Unit of : radian( rad) 弧度Always: =(t)
o
sr
x
s
Angular displacement: a body that rotates about a rotation axis, changing its angular position from 1 to 2 , undergoes an angular displacement:
12
Where is positive for counterclockwise rotation and negative for clockwise rotation.
Angular velocity and speed:
tavg
dt
d
Average angular velocity:
(Instantaneous) angular velocity:
avg and are vectors, with direction
s given by the right-hand rule. They are positive for counter-clockwise rotation and negative for clockwise rotation.
The magnitude of the body’s angular velocity is the angular speed.
Angular acceleration:
Average angular acceleration:
(Instantaneous) angular acceleration:
tavg
dt
d
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.
)(
)(
020
2
221
0
0
2
tt
t
See page 221: Table 11-1 equations of motion for constant linear acceleration and for constant angular acceleration
v
θθR
x
ΔS
0ω,
Δ
dt
ds
t
sv
t
0
lim
(1) Circular motion: description by linear variables
Speed:
Distance: S=r or s=r
11-5 relating the linear and angular variables
O X
R
v t( )
v t t( )
v t( )
v t t( )
v
tv
nv
切向 t 内法向 n
nata
nt
vt
t
vt
va
nt
nt
ˆˆ
ˆˆ
r
v
AB
vn
AB
)()( tvttvv t
v t( )
v t t( )
v
tv
nv r
ABvvn
r
v
t
AB
r
v
t
va
t
n
tn
2
00 limlim
dt
dv
t
va t
tt
0
lim
Radial acceleration:
Tangential acceleration:
rs rrdt
d
dt
dsv
rrdt
d
dt
dvat
rr
van
22
(2) Relating the linear and angular variables
Home work: 5E, 29p
11-6, 11-7 kinetic energy of rotation and rotational inertia
2
2
1 IK
(1)rotational kinetic energy:
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,
(Compare with kinetic energy of particle)
2222 )(2
1)(
2
1
2
1 iiiiii rmrmvmK
Let: 2iirmI
2
2
1 IK Then:
I is rotational inertia (or moment of inertia)(转动惯量)
O
ri
z
mi
vi
(2) Rotational inertia :
dmrI 2
2iirmI
For rigid body with continuously distributed mass:
Where r and ri represent the perpendicular distance from the axis of rotation to each mass element in the body.
dm
rM
For particle: I = m r 2
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:
2MhII com
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.
C hM
ICom I
平行
Proof of the parallel-axis theorem: (by yourself)
Question: What does the rotational inertia relate to?
Answer: Mass and its distribution.
11-8 torque
Fr
trFrF sin O ×
z
F
P
r
Unit: N.m
Where Ft is the component of perpendicular to , and is the angle between and .
Fr
Fr
Torque is a turning or twisting action on a body about a rotation axis due to a force.
(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.
Caution:
11-9 Newton’s second law for rotation (转动定律)
Compare with: amF
We obtain:
For rotation about fixed axis: Inet
Inet
Proof of equation:
f i
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:
amfF iii
Then:tiitit amfF
O
ω,α
ri
z
F i
miΔ
v
2iiitiiitiit rmramrfrF
For whole rigid body:
2iiiitiit rmrfrF
Ii 0
That is: Inet Inet
Sample problem: 11-7, 11-8
11-10 Work and rotational kinetic energy
From: rdFdW
For rigid body:
ddrFdsFdW tt
From initial angular position to final angular position, the work is:
dW 0
(1) work:
Caution:
(1) If torque is constant, then:
(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:
kKK
IIdI
ddt
dIdIdW
if
if
f
i
22
2
1
2
1
W
(4) The gravitational potential energy of rigid body:
cMghU
(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.
(3) power:
dt
d
dt
dWP
Sample problem: 11-9, 11-10
Home work: 29p, 65p, 66p
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