《理论力学》课程教学资源（PPT课件，英文）Chapter 10 Basic equations of particle dynamics

SITYOTHEORY MECHANICS1902Theoretical MechanicsCollege of Mechanicaland VehicleEngineeringWangXiaojun

Theoretical Mechanics THEORY MECHANICS College of Mechanical and Vehicle Engineering Wang Xiaojun

Part3DynamicsIntroductionDynamics studies the relationship between the mechanicalmotion ofan objectanditsforce.Mechanicalmodel:particleand particle system(includingrigid body)Two kinds of problems in dynamics:(1) Given the law of motion ofthe object, find the force acting on the object;(2) Given the force acting on the object and the initial conditions of motion, findthemotionlawoftheobject

Part3 Dynamics Introduction Dynamics studies the relationship between the mechanical motion of an object and its force. Mechanical model: particle and particle system (including rigid body). Two kinds of problems in dynamics: （1）Given the law of motion of the object, find the force acting on the object; （2）Given the force acting on the object and the initial conditions of motion, find the motion law of the object

Chapter10Basic equations ofparticle dynamicsFundamental lawofdynamicsDifferentialequations of particlemotionTwokindsofproblemsinparticledynamics

Chapter10 Basic equations of particle dynamics 引 言 • Fundamental law of dynamics • Differential equations of particle motion • Two kinds of problems in particle dynamics

10.1 FundamentallawofdynamicsFirst Law (Law ofinertia)If no force is exerted on any particle, it will remain in its original state of static oruniform motion in a straight lineThe property that a particle keeps its original motion state unchanged is called inertiaIn fact, there is no particle under no force, and if the force system acting on theparticle is the equilibrium force system, it is equivalent to the particle under no forceThe law states that force is what changes the motion of a particleSecondLaw (Law ofrelationshipbetweenforce and acceleration)The acceleration obtained by a particle underforce is directly proportionaltothe force force and inversely proportional to the mass ofthe particle. Thedirectionof accelerationis the same as the directionof force

First Law (Law of inertia) If no force is exerted on any particle, it will remain in its original state of static or uniform motion in a straight line. The property that a particle keeps its original motion state unchanged is called inertia. In fact, there is no particle under no force, and if the force system acting on the particle is the equilibrium force system, it is equivalent to the particle under no force. The law states that force is what changes the motion of a particle. Second Law (Law of relationship between force and acceleration) The acceleration obtained by a particle under force is directly proportional to the force force and inversely proportional to the mass of the particle. The direction of acceleration is the same as the direction of force. 10.1 Fundamental law of dynamics

10.1 Fundamental law of dynamicsFor ma= Fa =mThe above equation is the starting particle for deriving other dynamic equations, which iscalled the fundamental equation of dynamics.When aparticle is acted upon by several forcesat the same time, the above formula F should be understood as the resultant of these forces.parity conservation:1、 The relationship between force and acceleration is instantaneous, that is, the changeofforcetotheparticlemotionstateinacertaininstantisreflectedbytheaccelerationdetermined in that instant. The force does not directly determine the velocity of the particleandthedirectionofthevelocity canbe completely differentfromthedirectionoftheforce2、 If two equal forces act on two particles with different masses, the larger the mass is, thesmaller the acceleration will be.The smaller the mass, the greater the acceleration

m F a   = or ma F   = The above equation is the starting particle for deriving other dynamic equations, which is called the fundamental equation of dynamics. When a particle is acted upon by several forces at the same time, the above formula should be understood as the resultant of these forces. parity conservation： 1、The relationship between force and acceleration is instantaneous, that is, the change of force to the particle motion state in a certain instant is reflected by the acceleration determined in that instant. The force does not directly determine the velocity of the particle, and the direction of the velocity can be completely different from the direction of the force. 2、If two equal forces act on two particles with different masses, the larger the mass is, the smaller the acceleration will be. The smaller the mass, the greater the acceleration. F  10.1 Fundamental law of dynamics

10.1 FundamentallawofdynamicsThis shows: the greater the mass, the stronger the ability to maintain its original motion state,namely the greater the mass, the greater the inertia. Therefore, mass is a measure of theinertia of a particle.In a gravitational field, all objects are subjected to the force of gravity. The acceleration of anobject in free fall under theforce of gravity is called the acceleration of gravity and isdenotedbyg.ItcomesfromthesecondlawGG=mgm =gWhere, G is the magnitude of the force of gravity on the object, which is called the weight ofthe object, and g is the magnitude of the acceleration of gravity. Usually take g = 9.8%,In the SI system of units, the units of length, mass, and time are the basic units, in meters,kilograms, and seconds. The units of force are derived units, newtons. That is:1(N) = 1(Kg)×1(m/ s2)

This shows: the greater the mass, the stronger the ability to maintain its original motion state, namely the greater the mass, the greater the inertia. Therefore, mass is a measure of the inertia of a particle. In a gravitational field, all objects are subjected to the force of gravity. The acceleration of an object in free fall under the force of gravity is called the acceleration of gravity and is denoted by . It comes from the second law G mg   = g G m = Where, is the magnitude of the force of gravity on the object, which is called the weight of the object, and is the magnitude of the acceleration of gravity. Usually take . In the SI system of units, the units of length, mass, and time are the basic units, in meters, kilograms, and seconds. The units of force are derived units, newtons. That is: G g 9.8 2 s g = m 1( ) 1( ) 1( ) 2 N = Kg  m s g 10.1 Fundamental law of dynamics

10.1FundamentallawofdynamicsItmustbeparticleedoutthattheforceonaparticlehasnothingtodowithcoordinates.butthe acceleration of a particle has something to do with the choice of coordinates, soNewton'sfirst and second laws donotapplytoany coordinates.The coordinate system inwhich Newton's laws apply is called the inertial coordinate system.Otherwise, it is noninertialcoordinatesystemThird Law (Law of Action and Reaction)The acting force and the reacting force between two bodies are always equal and oppositeacting on the two bodies simultaneously along the same action lineThetheoryofmechanicsformedonthebasisofNewton'slawsiscalledclassicalmechanics

It must be particleed out that the force on a particle has nothing to do with coordinates, but the acceleration of a particle has something to do with the choice of coordinates, so Newton's first and second laws do not apply to any coordinates. The coordinate system in which Newton's laws apply is called the inertial coordinate system. Otherwise, it is non￾inertial coordinate system. Third Law (Law of Action and Reaction) The acting force and the reacting force between two bodies are always equal and opposite, acting on the two bodies simultaneously along the same action line. The theory of mechanics formed on the basis of Newton's laws is called classical mechanics. 10.1 Fundamental law of dynamics

10.2Differentialequation of particlemotion1、 Differential equations of particle motion in vector formAccording to the basic equationof dynamicsma=Fd?rdiKinematics showsthat:adt?dtdv-YThen we can get:dFmormdt?dt2Differential equations of particle motionin rectangular coordinates2XFFTmmmxd?dt?2Zdt

1、Differential equations of particle motion in vector form According to the basic equation of dynamics: Kinematics shows that: ma F   = 2 2 dt d r dt dv a    = = Then we can get: F dt dv m   = F dt d r m   = 2 2 or 2、Differential equations of particle motion in rectangular coordinates 2 2 x d x m F dt = 2 2 y d y m F dt = 2 2 z d z m F dt = 10.2 Differential equation of particle motion

10.2Differentialeguationsofparticlemotion3,DifferentialeguationsofparticlemotioninnaturalcoordinatesdvVF=F0=F,mmdtp2.SC0=FFFmmor1dipThis is the differentialequation forthe motion of a particlein naturalcoordinates

3、Differential equations of particle motion in natural coordinates F dt dv m = Fn v m =  2 = Fb 0 or F dt d s m = 2 2 Fn s m =  2  = Fb 0 This is the differential equation for the motion of a particle in natural coordinates. 10.2 Differential equations of particle motion

10.3TwokindsofproblemsinparticledynamicsThe first type of problem: given the motion of a particle, find the force actingon it. The essence of this kind of problem boils down to a mathematical problemofderivation.The second kind of problem: given the force acting on the particle, to find themotion of the mass. The essence of this kind of problem can be reduced to solvingdifferential equation or integral problem in mathematics1、When the force is a constantor a simple functionofdv=F(t), so ("mdv= ['F(t)dt 。time,dt

The first type of problem: given the motion of a particle, find the force acting on it. The essence of this kind of problem boils down to a mathematical problem of derivation. The second kind of problem: given the force acting on the particle, to find the motion of the mass. The essence of this kind of problem can be reduced to solving differential equation or integral problem in mathematics. 1、When the force is a constant or a simple function of time, m F t dv ( ) ，so 。 dt = ( ) 0 0 v t v mdv F t dt =   10.3 Two kinds of problems in particle dynamics