物理雙語教學Chapter 6 Rotation 定軸轉(zhuǎn)動

1、Chapter 6 RotationIn 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

2、 bowling ball rolling down a bowling lane. Such a ball is in rolling motion, rotating about a moving axis.6.1 The Rotational Variables1. 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

3、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

4、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

5、 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 posit

6、ive direction of the x axis, and is given by 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

7、 have relations 4. If the body rotates about the rotation axis as in the right figure, changing the angular position of the reference line from to , the body undergoes an angular displacement given by The definition of angular displacement holds not only for the rigid body as a whole but also for ev

8、ery 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 pos

9、ition at time and at angular position at time . We define the average angular velocity of the body in the time interval from to to be In which is the angular displacement that occurs during . (2). The (instantaneous) angular velocity , with which we shall be most concerned, is the limit of the avera

10、ge angular velocity as is made to approach zero. Thus If we know , we can find the angular velocity by 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 sp

11、eed, 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 th

12、e 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 and be the angular velocity at times and , respectively. The average angular acceleration of the rotating body in the inte

13、rval from to is defined as In which is the change in the angular velocity that occurs during the time interval . (2). The (instantaneous) angular acceleration , with which we shall be most concerned, is the limit of this quantity as is made to approach zero. Thus above equations hold not only for th

14、e 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 th

15、e angular velocity. 7. Rotation with constant angular acceleration: Here we suppose that at time . 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: Ang

16、ular velocity： Angular acceleration：6.2 Kinetic Energy of Rotation1. To discuss kinetic energy of a rigid body, we cannot use the familiar formula directly because it applies only to particles. Instead, we shall treat the object as a collection of particles-all with different speeds. We can then add

17、 up the kinetic energies of these particles to find kinetic energy of the body as a whole. In this way we obtain, for the kinetic energy of a rotating body, In which is the mass of the ith particle and is its speed. The sum is taken over all the particles in the body.2. The problem with above equati

18、on is that is not the same for all particles. We solve this problem by substituting for in the equation with , so that we have In which is the same for all particles.3. The quantity in parentheses on the right side of above equation tells us how the mass of the rotating body is distributed about its

19、 axis of rotation. (1). We call that quantity the rotational inertia (or moment of inertia) I of the body with respect to the axis of rotation. Its a constant for a particular rigid body and for a particular rotation axis. We may now write (2). The SI unit for I is the kilogram-square meter ().(3).

20、The rotational inertia of a rotating body depends not only on its mass but also on how that mass is distributed with respect to the rotation axis. 4. We can rewrite the kinetic energy for the rotating object as Which gives the kinetic energy of a rigid body in pure rotation. Its the angular equivale

21、nt of the formula , which gives the kinetic energy of a rigid body in pure translation.6.3 Calculating the Rotational Inertia1. If a rigid body is made up of discrete particles, we can calculate its rotational inertia from . 2. If the body is continuous, we can replace the sum in the equation with a

22、n integral, and the definition of rotational inertia becomes . In general, the rotational inertia of any rigid body with respect to a rotation axis depends on (1). The shape of the body, (2). The perpendicular distance from the axis to the bodys center of mass, and (3). The orientation of the body w

23、ith respect to the axis.The table gives the rotational inertias of several common bodies, about various axes. Note how the distribution of mass relative to the rotational axis affects the value of the rotational inertia I. We would like to give the example of rotational inertia for a thin circular p

24、late a thin rod 3. The parallel-axis theorem: If you know the rotational inertia of a body about any axis that passes through its center of mass, you can find its rotational inertia about any other axis parallel to that axis with the parallel-axis theorem: Here M is the mass of the body and h is the

25、 perpendicular distance between the two parallel axes. 4. Proof of the parallel-axis theorem: Let O be the center of mass of the arbitrarily shaped body shown in cross section in the figure. Place the origin of coordinates at O. Consider an axis through O perpendicular to the plane of the figure, an

26、d another axis of P parallel to the first axis, Let the coordinates of P be a and b.Let dm be a mass element with coordinates x and y. The rotational inertia of the body about the axis through P is then Which we can rearrange as 6.4 Newtons Second Law for Rotation1. Torque: The following figure show

27、s a cross section of a body that is free to rotate about an axis passing through O and perpendicular to the cross section. A force F is applied at point P, whose position relative to O is defined by a position vector r. Vector F and r make an angle with each other. (For simplicity, we consider only

28、forces that have no component parallel to the rotation axis: thus, F is in the plane of the page). We define the torque as a vector cross product of the position vector and the force Discuss the direction and the magnitude of the torque.2. Newtons second law for rotation(1). The figure shows a simpl

29、e case of rotation about a fixed axis. The rotating rigid body consists of a single particle of mass m fastened to the end of a massless rod of length r. A force F acts as shown, causing the particle to move in a circle about the axis. The particle has a tangential component of acceleration governed

30、 by Newtons second law: . The torque acting on the particle is . The quantity in parentheses on the right side of above equation is the rotation inertia of the particle about the rotation axis. So the equation can be reduced to . (2) For the situation in which more than one force is applied to the p

31、article, we can extend the equation as . Where is the net torque (the sum of all external torques) acting on the particle. The above equation is the angular form of Newtons second law. (3) Although we derive the angular form of Newtons second law for the special case of a single particle rotating about a fixed axis, it holds for any rigid body rotating about a fixed axis, because any such body can be analyzed as an assembly of single particles.6.5 Work and Rotational Kinetic Energy1. Work-k