英文版大学物理 第五章

(radian measure)
Equation of motion for a rotating body: = (t)
2. Angular Displacement When the body rotates about the rotation axis from 1 to 2, the body undergoes an angular displacement y 2 1 Reference At t2 line An angular displacement in the counterclockwise direction is positive, in the clockwise direction is negative. At t1 1 2 x 3. Angular Velocity O Rotation axis Average angular velocity The angular velocity is either 2 1 avg positive or negative, depending t2 t1 t on whether the body is rotating Angular velocity counterclockwise (positive) or d lim clockwise (negative). t 0 t dt
0
0 0 t t
1 2
2
x x0 v0t 1 at 2 2
2 v 2 v0 2a( x x0 )
2 2 0 2 ( 0 )
avg
0
2
vavg
v v0 2
Relating with Linear and Angular Variables
avg 2 1
t 2 t1 t
The (instantaneous) angular acceleration ,, is the limit in this equation as t approaches zero.
d lim t 0 t dt
z
Rotation axis
y Body Reference line y
r s
O
Rotation axis
x
O
x
Zero angular position
1. Angular Position A reference line fixed in the body, perpendicular to the rotation axis, and rotating with the body. The angular position of reference line describes the angular position of the body:
The Position s=r
y
The Speed Differentiating the equation with respect to time—with r held constant
ds d r dt dt
r s
O
Rotation axis
x
v r
Zero angular position
5-2 Kinetic Energy of Rotation A rigid body is treated as a collection of particles. z Kinetic energy of a rotating rigid body,
K m v m v
1 mv 2
4. Angular Acceleration
Let 2 and 1 be its angular velocities at time t2 and t1, respectively. The average angular acceleration of the rotating body in the interval from t1 to t2 is
Linear mass density Surface mass density Volume mass density
Rotational inertia depends on following factors: Mass and shape of rigid body , position of rotation axis
= 0 +t
at r

0
t
r
2.56 0 (15) 0.38 m / s 2 0.039g 100
Although the final radial acceleration ar = 10g is large (and alarming), the astronaut’s tangential acceleration at during the speed-up is not.
z Axis z
Axis
Right-hand rule z Axis Speeding up Axis Slowing down z
5. Rotation with Constant Angular Acceleration In pure translation, motion with a constant linear acceleration is an important special case. In pure rotation, the case of constant angular acceleration is also important, and a parallel set of equations holds for this case also. Angular Linear 0 t v v at
at = r = 0 Therefore a = ar = 2 r = 10g
ar (10)(9.8) 2.56 rad / s 24 rev / s r 15
(Answer)
(b) What is the tangential acceleration of the astronaut if the centrifuge accelerates at a constant rate from rest to the angular speeds of (a) in 100 s? Solution: Because the angular acceleration is constant Then
Checkpoint 1 The Acceleration Differentiating the equation with respect to time—again with r held constant dv d v2 2
dt dt r
at r
ar
r
r
Example 5-1 A centrifuge is used to accustom astronauts to the large acceleration experienced during a liftoff. The radius r of the circle traveled by an astronaut is 15 m. (a) At what constant angular speed must the centrifuge rotate if the astronaut is to have a linear acceleration of magnitude of 10g? Solution: Because = constant, d/dt = = 0 ,
Example 5-2 Show that the rotational inertia of a uniform annular cylinder (or ring) of inner radius R1, outer radius R2, and mass M, is I 1 M( R12 R22 ) , 2 as stated in Table 5-1b, if the rotation axis is through the center along the axis of symmetry. Divide the cylinder into thin concentric Solution： cylindrical rings or hoops of thickness dr, each with mass R1 r dm dV dr is the mass density of the body. or
Chapter 5 Rotations of Rigid Bodies
5-1 The Kinematics of Rotations about a Fixed Axis 5-2 Kinetic Energy of Rotation 5-3 Calculating the Rotational Inertia 5-4 Torque 5-5 Newton’s Second Law for Rotation 5-6 Work and Rotational Kinetic Energy 5-7 Angular Momentum 5-8 Newton’s Second Law in Angular Form 5-9 Conservation of Angular Momentum 5-10 The Spinning Top
1 2 2 1 1 1 百度文库 2 2 2
2 i i
o
ri
mi
vi
mi is the mass of the ith particle and vi is its speed.
K 1 mi vi2 1 ( mi ri2 ) 2 2 2
rotational inertia (or moment of inertia) I Kinetic Energy of Rotation
m r
2
i i
K 1 I 2 2
5-3. Calculating the Rotational Inertia If a rigid body consists of a few particles
If a rigid body consists of a great many of particles(it is continuous) Linear distribution: Surface distribution: Volume distribution:
The unit of angular acceleration is commonly the radian per second-squared (rad/s2) or the revolution per second-squared (rev/s2).
Are Angular Quantities Vectors?
5-1 The Kinematics of Rotations about a Fixed Axis A rigid body is a body that can rotate with all its parts locked together and without any change in its shape. A fixed axis means that the rotation occurs about an axis that does not move. z In the description of the rotational Rotation Body motion, we shall introduce angular axis equivalents of the linear quantities position, displacement, velocity, r P and acceleration. y These quantities not only describe x O the rigid body as a whole but also for every particle within that body.