Chapter 4 Rotation of rigid body §4.1 The rotation of a rigid body about a fixed axisThe rotation of a rigid body about a fixed axis §4.2 Torque, the law of rotation, moment of inertiaTorque, the law of rotation, moment of inertia §4.3 Angular momentum the law of angular momentum conservationAngular momentum the law of angular momentum conservation §4.4 work done by a torque, the theorem of kinetic energy of a rigid body rotating about a fixed axiswork done by a torque, the theorem of kinetic energy of a rigid body rotating about a fixed axis Summary

§4.1 The rotation of a rigid body about a fixed axis 1. the angular velocity and angular acceleration of a rotating rigid bodythe angular velocity and angular acceleration of a rotating rigid body 2. Formulation of fixed axis rotation with constant angular accelerationFormulation of fixed axis rotation with constant angular acceleration 3. The relationship between angular quantities and linear quantitiesThe relationship between angular quantities and linear quantities

Rigid body ： under external forces if the shape and size of an object do not change ． ( the distance between any two arbitrary points in the object is a constant ) The motion of a rigid body ： the translation rotation ． ⑴ Rigid body is an idealized object ⑵ The model of the rigid body is introduced for trying to simplify the study of the motion. explain ：

Axis Rigid body 1. Angular coordinate Reference plane which is perpendicular to the Oz axis ） (t)(t) (t+△t)(t+△t) Reference direction An arbitray point in Equation motion of rotating rigid body about a fixed axis 2. Angular displacement 3. Angular velocity Rotation clockwise rotation counter clockwise  0  0   direction of : determined by right hand rule 1 the angular velocity and angular acceleration of a rotating rigid body

4. angular acceleration The rotation direction can be expressed by the positive or negative of the angular velocity, when the rigid body rotating about a fixed axis (i) When  >0, the rigid body rotates with acceleration;. (ii) When  <0, the rigid body rotates with deceleration. z z (i) When rotating counter clockwise,  > 0; (ii) When rotating clockwise,  < 0; When  > 0 1 the angular velocity and angular acceleration of a rotating rigid body

( 1 ) The location and direction of rotation axis are fixed relative to a inertial reference frame. ( 3 ) Every point moves through the same angle during a particular time interval ； The Features of Rotation about a Fixed Axis: ( 4 ) Each rotation planeis perpendicular to rotation axis. ． ( 2 ) Every point of body moves in a circle whose center lies on rotation axis and radius different.

2 Formula of fixed axis rotation with constant angular acceleration Fixed axis rotation constant angular acceleration Straight line motion with constant linear acceleration A rotation with variable angular velocity is called fixed axis rotation with constant angular acceleration ．

3 The relationship between angular quantities and linear quantities P

Example 1 In a mini-motor rotates in a high speed, in which a cylindrical rotor rotates about an axis that is perpendicular to the cross-section and passed through is center. Initially, the angular speed of the mini-motor is zero then it up in a time relationship of, where ． ( 1) what is the rotate speed of the mini-motor at t =6 s. (2)how many turns it has made in the time interval of at t =6 s ． (3)what is the discipline of the angular acceleration varying with respect of time ．

(2) The turns the motor has made in the time interval of t=6 s is Solution: ( 1 ) substitute t=6 s to ( 3 ) The angular acceleration of the motor is

Example 2 An electric motor rotates with a high speed, in which a cylindrical rotor can rotate about the axis going through its center and perpendicular to the cross – section area of the rotator. Initially, the angular velocity is. After 300 s the speed reached r/min. it is known that the angular acceleration a of the rotation is proportional to time. How many revolutions has the turned in this time interval ？ Solution : ， ， integrating both side 得

at t =300 s

from We have The rotator has made the following revolutions in 300 s

§4.2 Torque, the law of rotation, moment of inertia 1. TorqueTorque 2. The law of rotationThe law of rotation 3. The power of torqueThe power of torque 4. The theorem of parallel axesThe theorem of parallel axes

P O Is the arm of force the torque of force with respect to axis z is 1 Torque 1 Torque Describe the effect of force on the rotation of a rigid body ． * When the combined force is zero, their combined torque may not be zero Right hand rule

O （ 1 ） If the force is not in the rotation plane, it can be decomposed into two components with parallel and normal to the rotation axis. where the torque of is zero, so the torque of is discussion

方向 the torque of the external forces drives the plate to rotate M = r × F torque Tangential component 1 F  F r M 叉乘右螺旋 1 M 2 M M = r × F M = r F sin  Magnitude 2 r 2 = 2 F  d 2 = 2 F 1 M 2 M Total internal torque = M + d 2 2 F Magnitude M = d 1 1 F = r 2 2 F  r 1 1 F  r 1 = 1 F  M = r F sin  Magnitude 1 d 1 = 1 F  1 d 1 F 2 r 2 2 F  P 2  2 d 2 Tangential component （2）（2） External torque and total external torque r 1 F 1 P 1 O

The total external torque is equal to the algebraic sum of these external torque （ 3 ） The combined torque generated by the internal forces between the mass points in a rigid body is 0, i.e. O O (4) The torque is zero, when the force acting on the axis

when the rigid body rotates about a fixed axis, the angular acceleration is proportional to the combined external torque that the rigid body is subject to, and it is inversely proportional to the moment of inertia of the rigid body. The law of rotation Moment of inertia O

（2）（2） （3）（3） （ 1 ） is a constant The law of rotation discussion

Exercises p.144 / 4- 6, 7, 9

§4.3 Angular momentum, the law of angular momentum conservation 1. The theorem of angular momentum and the law of the conservation of the angular momentum of mass pointsThe theorem of angular momentum and the law of the conservation of the angular momentum of mass points 2. The theorem of angular monentum and the priciple of conservation of angular momentum of a rigid body rotatingThe theorem of angular monentum and the priciple of conservation of angular momentum of a rigid body rotating

The accumulation effect of forces over time ： Impulse 、 Momentum 、 the theorem of momentum ． The accumulation effect of Torque over time ： Impulse torque 、 Angular momentum 、 the theorem of angular momentum of a rigid body rotating about a fixed axis ． §4.3 Angular momentum, the law of angular momentum conservation

1 The theorem of angular momentum and the law of the conservation of the angular momentum of mass points Description for the motion of a mass point Can we use the momentum to descript the motion of a rigid body ? “momentum ” is not good physical q to describe to the rotation of the rigid body.

so and

Exercises p.144 / 4- 13, 19, 28

§4.4 Work Done by a Torque 1. Work done by a torqueWork done by a torque 2. The power of a torqueThe power of a torque 3. The kinetic energy of rotationThe kinetic energy of rotation 4. The theorem of kinetic energy of a rigid body rotating about a fixed axisThe theorem of kinetic energy of a rigid body rotating about a fixed axis

The accumulation effect of forces over space ： Work 、 Kinetic energy 、 the theorem of Kinetic energy ． The accumulation effect of Torque over space ： Work done by a torque 、 the kinetic energy of rotation 、 the theorem of Kinetic energy of a rigid body rotating about a fixed axis ．

Work done by a torque: 1 Work done by a Torque 2 The power of a torque Comparing to

For a mass element the line sped is The kinetic energy of the mass element is ∑ The kinetic energy of the entire rigid body is ∑ Moment of inertia J J We have 3 The kinetic energy of rotation

The theorem of Kinetic energy of a mass point: The theorem of Kinetic energy of a rigid body rotating: From the element of work done by the external torque The law of rotation so The work done by the combined external torque The increment of the rotation kinetic energy of the rigid body 4 The theorem of kinetic energy of a rigid body rotating about a fixed axis

Example 1 The turnplate of a gramophone rotates in an angular velocity about the axis which goes through the center of the plate. After a record being put on it, the record will rotate will rotate with the turnplate under the action of friction force. Assume the radius of the plate is R and the mass is m ， the friction factor is.(1 ) what is the magnitude of the torque of the friction force ； ( 2 ) how long does the record need when its angular velocity reaches ； ( 3 ) what is the work done by the drive force of the turn plate in this period of time ？

R o Solution (1) as shown in figure, an element area, ds = drdl ， the friction force of which the element subjected is The torque of the friction force to the point O on the rotating axis of the trunplate is r drdr dldl

for the cirque with a width dr, the torque of friction force that the turnplate subject is R r drdr dldl o

(3) From the angle that the record turns in the time interval of 0 to t is ( 2 ) According to the theorem of angular momentum is (the moment of inertia of record is J=mR 2 /2 ) （ the record makes a rotation motion with a uniform acceleration ） The word done by the drive force of the turnplate in this time interval is Based on

Example 2 A rod of length is l and mass m’ can rotate about pivot O freely, a bullet with mass m and speed v is shot into the point with distance a away from the pivot, rendering a 30 o. Deflection angle of the rod with respect to the vertical axis, what is the initial speed of the? Solution Take the bullet and the rod as a system, the angular momentum of the system should be conserved

After the bullet gets into the rod, taking the bullet, the rod and the earth as a system, the mechanical momentum of the system should be conserved ， E = constant ．

Exercises p.147 / 4- 30, 31, 36

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