《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

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

θ is measured relative to the positive direction of the x axis, and θ is given by ).(measure radian r s

=θ

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

rad r

r rev o ππ223601=== 4. If the body rotates about the rotation

axis as in the right figure, changing

the angular position of the reference

line from 1θ to

2θ, the body undergoes an angular displacement

θ∆ given by 12θθθ-=∆

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 θ∆ of a rotating body can be either positive or negative, depending on whether the body is rotating in the direction of increasing θ (counterclockwise ) or decreasing θ (clockwise ).

5. Angular velocity

(1). Suppose that our rotating body is at angular position 1θ at time 1t and at angular position 2θ at time 2t . We define

the average angular velocity of the body in the time interval t ∆ from 1t to 2t to be

t t t ∆∆=--=θθθω121

2 In which

θ∆ is the angular displacement that occurs during t ∆.

(2). The (instantaneous) angular velocity ω, with which we shall be most concerned, is the limit of the average angular velocity as t ∆ is made to approach zero. Thus

dt d t t θθω=∆∆=→∆0lim If we know

)(t θ, we can find the angular velocity ω by