力学专业英语_8.pdf

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1、Lesson 9 Theorems of particle dynamics In solving many problems of dynamics it will be found that the general theorems of particle dynamics are more conveniently applied than the method of integration of differential equations of motion.The importance of the general theorems is that they establish v

2、isual relationships between the principal dynamic characteristics of motion of material bodies,thereby presenting broad possibilities for analyzing the mechanical motions widely employed in practical engineering.Furthermore,the general theorems make it possible to study for practical purpose the spe

3、cific aspects of a given phenomenon without investigating the phenomenon as whole.Now let us see how these theorems apply to one material particle.The vector approach will now be used in order that the motion may be discussed without the introduction of a specific coordinate system.As shown in Figur

4、e 1,a particle P is located at the point with position vector r relative to a given origin o.The velocity and acceleration of the particle relative to o are v=r and a=r respectively.If the particle is of mass m and is moving under the action of a force F then Newtons second law of motion leads to th

5、e equation motion as follows,Mr=F.If more than one force is acting on the particle,then F is the vector sum of all the individual force.It will always be assumed that the mass m of a particle is a constant.This means that Eq.(1)can be expressed by the following equation.the vector mv is defined to b

6、e the linear momentum of the particle Eq.(2)shows that the rate of change of the linear momentum is equal to the total force acting on the particle.If the total force F acting on a given particle is zero then it follows from Eq.(2)that the linear momentum mv is a constant.The linear momentum is said

7、 to be conserved and the equation mv=constant is called the equation of conservation of linear momentum.Let a particle of mass m moving under the action of a force F have a velocity VO at time t=tO,and at time t1 let its velocity be V1.Now multiply both sides of Eq.(2)by dt and take definite integra

8、ls,we have.As shown in Eq.(3),the integral on the right side is the impulse of the acting force during a certain interval of time.In words,Eq.(3)states that the change in the linear momentum equals the linear impulse.Often,in analyzing the motion of a particle,ti is necessary to consider the change

9、not in the vector mv itself but in its moment.Taking the vector product of Eq.(2)with r gives.The expression is called the moment of the linear momentum or angular momentum of the particle about point o.Eq.(5)shows that the rate of change of the angular momentum of the particle about o is equal to t

10、he moment about o of the total force acting on the particle.If the moment rF is zero,then we have rmv=constant.That is the equation of conservation of angular momentum about point o.Now consider a particle which travels from a point with position vector r to another point with position vector r+r un

11、der the action of a non-constant force F.The force will remain almost constant and so the infinitesimal amount or work w done by this force will be defined by w=F r.The total work done by the force in moving the particle from some fixed position rO to its current position r is thereforr defined as W

12、=F dr.In general,the value of this integral will depend upon the path which the particle takes in moving from rO to r.If in fact the integral is independent of the path,then the force F is defined to be conservative.For such a force,the integral will define a unique function of position.For any vect

13、or v,and according to the chain rule for differentiation the above equation for the work then becomes.The quantity mv2/2 is defined to be the kinetic energy of the particle.In words,the work done by the forces acting on a particle is the amount by which the kinetic energy of the particle changes from the initial to the final position.