《物理双语教学课件》Chapter 6 Rotation 定轴转动

1 Chapter 6 Rotation

In this chapter, we deal with the rotation of a rigid body

about a fixed axis. The first of these restrictions means that we

shall not examine the rotation of such objects as the Sun,

because the Sun-a ball of gas-is not a rigid body. Our second

restriction rules out objects like a bowling ball rolling down a

bowling lane. Such a ball is in rolling motion, rotating about a

moving axis.

6.1 The Rotational Variables

1. Translation and Rotation: The motion is the one of pure

translation, if the line connecting any two points in the object

is always parallel with each other during its motion.

Otherwise, the motion is that of rotation. Rotation is the

motion of wheels, gears, motors, the hand of clocks, the rotors

of jet engines, and the blades of helicopters.

2. The nature of pure rotation: The

right figure shows a rigid body of

arbitrary shape in pure rotation

around a fixed axis, called the axis

of rotation or the rotation axis. (1). Every point of the body moves in a circle whose center 2 lies on the axis of the rotation.

(2). Every point moves through the same angle during a

particular time interval.

3. Angular position: The above figure shows a reference line,

fixed in the body, perpendicular to the axis, and rotating with

the body. We can describe the motion

of the rotating body by specifying the

angular position of this line, that is,

the angle of the line relative to a fixed

direction. In the right figure, the

angular position  is measured relative to the positive

direction of the x axis, and  is given by

).(measureradianrs

Here s is the length of the arc (or the arc distance) along a

circle and between the x axis and the reference line, and r is a

radius of that circle.

An angle defined in this way is measured in radians (rad)

rather than in revolutions (rev) or degree. They have relations

radrrrevo223601

4. If the body rotates about the rotation

axis as in the right figure, changing

the angular position of the reference 3 line from 1 to 2, the body undergoes an angular

displacement  given by

12

The definition of angular displacement holds not only for the

rigid body as a whole but also for every particle within the

body. The angular displacement  of a rotating body can

be either positive or negative, depending on whether the body

is rotating in the direction of increasing  (counterclockwise)

or decreasing  (clockwise).

5. Angular velocity

(1). Suppose that our rotating body is at angular position 1

at time 1t and at angular position 2 at time 2t. We define

the average angular velocity of the body in the time interval

t from 1t to 2t to be

ttt1212

In which  is the angular displacement that occurs during

t.

(2). The (instantaneous) angular velocity , with which we

shall be most concerned, is the limit of the average angular

velocity as t is made to approach zero. Thus

dtdtt0lim

If we know )(t, we can find the angular velocity  by 4 differentiation.

(3). The unit of angular velocity is commonly the radian per

second (rad/s) or the revolution per second (rev/s).

(4). The magnitude of an angular velocity is called the

angular speed, which is also represented with .

(5). We establish a

direction for the vector

of the angular velocity

 by using a

right-hand rule, as

shown in the figure.

Curl your right hand about the rotating record, your fingers

pointing in the direction of rotation. Your extended thumb

will then point in the direction of the angular velocity vector.

6. Angular acceleration

(1). If the angular velocity of a rotating body is not constant,

then the body has an angular acceleration. Let 2 and 1 be

the angular velocity at times 2t and 1t, respectively. The

average angular acceleration of the rotating body in the

interval from 1t to 2t is defined as

ttt1212

In which  is the change in the angular velocity that occurs 5 during the time interval t.

(2). The (instantaneous) angular acceleration , with which

we shall be most concerned, is the limit of this quantity as t

is made to approach zero. Thus

dtdtt0lim

above equations hold not only for the rotating rigid body as a

whole but also for every particle of that body.

(3). The unit of angular acceleration is commonly the radian

per second-squared (rad/s2) or the revolution per

second-squared (rev/s2).

(4). The angular acceleration also is a vector. Its direction

depends on the change of the angular velocity.

7. Rotation with constant angular acceleration:

2000021)(ttdttdtdtdtdtddtd

Here we suppose that at time ,0t 00. We also can get a

parallel set of equations to those for motion with a constant

linear acceleration.

8. Relating the linear and angular variables: They have relations

as follow:

Angular displacement: rdsdd

Angular velocity： rv