《物理双语教学课件》Chapter 6 Rotation 定轴转动

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

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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 ??sr? is measured relative to the positive

direction of the x axis, and ? is given by

(radianmeasure).

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 1rev?360o?2?r?2?rad r4. If the body rotates about the rotation axis as in the right figure, changing the angular position of the reference

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line from

?1 to

???2, the body undergoes an angular

displacement

given by

????2??1

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 t1 and at angular position ?2 at time t2. We define the average angular velocity of the body in the time interval

?t

from t1 to t2 to be

???2??1t2?t1????t In which

?t.

??

is the angular displacement that occurs during

(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

??lim??d???t?0?tdt If we know

?(t), we can find the angular velocity ? by

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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 rule,

a as

right-hand

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 the angular velocity at times

t2

?2

and

?1

be

and t1, respectively. The

average angular acceleration of the rotating body in the interval from t1 to t2 is defined as In which

???2??1t2?t1??? ?t?? is the change in the angular velocity that occurs

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