college of physics science & technology yangzhou universitychina chapter 11rotation 11.1 the...
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College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
Chapter 11 ROTATION
11.1 The Motion of Rigid Bodies
11.1.1 Rigid bodies
A rigid body is defined as a body of which the distance between any two particles is not changed.
This is another ideal model for dealing with physical problems.
College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
11.1.2 The classification of the motion of rigid bodies
(i) Translational motion: The motion in which any straight line in a rigid body is always parallel;
(ii) Rotation about a fixed axis: The motion in which any particle in a body is in a circular motion of a radius about an identical straight line keeping constant position;
(iii) Rotation about a fixed point;
(iv) A planar motion of a rigid body: The motion in which any particle in the body is always in its plane;
(v) A general motion of a rigid body.
College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
11.2 Angular Kinematics
12.2.1 Rotation about a fixed axis
In a rotation about a fixed axis, any particle in the body turns the same angle as others though the distance to the axis may be different. Therefore, we can chose an angular coordinate to represent the position of the rigid body.
Angular displacement: 12
Average angular velocity:t
Instantaneous angular velocity: dt
d
ttt
00
limlim
College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
Average angular acceleration: t
Instantaneous angular acceleration: 2
2
0lim
dt
d
dt
d
tt
On the other hand, with some initial conditions, angular equation can be solved:
2
1
2
1
t
tdtd
2
1
2
1
t
tdtd
College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
11.2.2 Relation of linear and angular quantities
The relationship between linear and angular quantities:
rv
2
ra
ra
n
Further definition of angular velocity vector:
dt
d
And angular acceleration vector:
dt
d
College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
Thus,
rv
)( ra
ra
n
College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
11.3 Kinetic Energy of Rotation
11.3.1 Kinetic energy of rotation
The kinetic energy of a rigid body is
222222
2
1
2
1)(
2
1
2
1 IrmrmvmK iiiiii
Rotational inertia: 2ii rmI
College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
11.3.2 Calculating the rotational inertia
For continuously distributed mass:
dVrI 2
For a collection of point masses :
2ii rmI
It is related to the mass distribution about an axis.
College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
Parallel-axis theorem:
2CM MdII
Normal-axis theorem:
yxz III
College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
11.4 Newton’s Second Law in Angular Form
sinrF
Fr
Generally, torque is defined as:
11.4.1 Torque
College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
The torque of a force about a axis is the component of the torque about a point on the axis.
College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
11.4.2 Newton’s second law for rotation
We consider the simple situation:
maF
The torque:
rmarF
)()( 2mrrrm
I
Generally, we have
Inet
College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
11.4.2 Newton’s second law for rotation
We consider the simple situation:
maF
The torque:
rmarF
)()( 2mrrrm
I
Generally, we have
Inet
College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
d
dI
dt
d
d
dI
dt
dI
From the rotation law:
dId
21
22 2
1
2
12
1
2
1
IIdIdW
We get the work-kinetic energy theorem for rotation
College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
21
2212 2
1
2
1 IIUU
If the force acting on the body is conservative, then we have
The conservation of mechanical energy in rotational motion:
][constant2
1 2 UIE
College of Physics Science & Technology
YANGZHOU UNIVERSITY CHINA
Problems:
1. 11-4 (on page 239), 2. 11-25, 3. 11-37, 4. 11-38, 5. 11-57, 6. 11-66.