Chapter 4 Rotation of rigid body4.1 The rotation of a rigid body about a fixed axis4.2 Torque , the law of rotation , moment of inertia4.3 Angular momentum the law of angular momentum conservation4.4 work 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 axis1. the angular velocity and angular acceleration of a rotating rigid body2. Formulation of fixed axis rotation with constant angular acceleration3. The relationship between angular quantities and linear quantities

Rigid bodyunder 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 bodythe translation rotation

1 the angular velocity and angular acceleration of a rotating rigid body

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;(ii) When rotating clockwise, < 0;When > 01 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. 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 accelerationStraight 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 quantitiesP

Example 1In 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 isSolution: (1) substitute t=6 s to (3) The angular acceleration of the motor is

Example 2An 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 18000 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

from

4.2 Torque, the law of rotation, moment of inertia1. Torque2. The law of rotation3. The power of torque4. The theorem of parallel axes

PO 1Torque 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

O1If the force is not in the rotation plane , it can be decomposed into two components with parallel and normal to the rotation axis.

2External torque and total external torque

The total external torque is equal to the algebraic sum of these external torque3The combined torque generated by the internal forces between the mass points in a rigid body is 0, i.e.(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.Moment of inertia

Exercisesp.144 / 4- 6, 7, 9

4.3 Angular momentum, the law of angular momentum conservation1. The theorem of angular momentum and the law of the conservation of the angular momentum of mass points2. The 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 momentumThe accumulation effect of Torque over time Impulse torque Angular momentum the theorem of angular momentum of a rigid body rotating about a fixed axis4.3 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 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 .

Exercisesp.144 / 4- 13, 19, 28

4.4 Work Done by a Torque1. Work done by a torque2. The power of a torque3. The kinetic energy of rotation4. The 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 torquethe 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

3The kinetic energy of rotation

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 mthe 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

Ro Solution (1) as shown in figure, an element area, ds = drdlthe 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

for the cirque with a width dr, the torque of friction force that the turnplate subject isRrdrdlo

(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=mR2/2) the record makes a rotation motion with a uniform acceleration

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 30o . Deflection angle of the rod with respect to the vertical axis, what is the initial speed of the?SolutionTake 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

Exercisesp.147 / 4- 30, 31, 36

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