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1、12.4 Theorem of motion of the center of mass,Chapter 12:Theorem of Momentum,12.1 The center of mass of a system of particles,12.2 Momentum and impulse,12.3 Theorem of momentum,The center of mass.The center of mass of a system of particles is called center of mass.It is an important concept represent
2、ing the distribution of mass in any system of particles.,The position of thecenter of mass c is,12.1 The Center of Mass of a System of Particles,External forces are the forces exerted on the members of a system by particles or bodies not belonging to the given system Internal forces are the forces o
3、f interaction between the members of the same system.As far as the whole system of particles is concerned,the geometrical sum(the principal vector)of all the internal forces of a system is zero.The sum of the moments(the principal moment)of all the internal forces of a system with respect to any cen
4、ter of axis is zero,too.,2.External forces and internal forces of a system of particles,1)Momentum of a particle.The product of the mass of a particle and its velocity is called the momentum of a particle.It is a time-dependent vector with the same direction as the velocity,the unit of which is kgm/
5、s.,Momentum is a physical quantity measuring the intensity of the mechanical motion of a material body.For example,the velocity of a bullet is big but its mass is small.In the case of a boat it is just opposite.,12.2 Momentum and Impulse,1.Momentum,2)The momentum of a system of particles is defined
6、as the vector equal to the geometric sum of the momenta of all the particles of the system:,The momentum of a system is equal to the product of the mass of the whole system and the velocity of its center of mass.,.,3)Momentum of a system of rigid bodies:Assume that the mass and the velocity of the c
7、enter of mass of the i-th rigid body are.For the whole system we get then,.,.,.,In terms of projections on cartesian axes we have,In the mechanism shown in the figure,OA rotate with a constant angular velocity w.Assume that OA=L,OB=L.The rods OA and AB are homogeneous and of mass m.The mass of the s
8、lide block at B is also m.Determine the momentum of the system when j=45.,Example,For the rod OA mass=.For the slide block mass=.For the rod AB:mass=,Solution:,2.Impulse:The product of a force and the action time of the force is called impulse.Impulse is used to characterize the accumulated effect o
9、n a body of a force acting during a certain time interval.,3)The impulse of a resultant force is equal to the geometric sumof the impulses of all component forces:,The unit of impulse is the same as that of momentum.,1.Theorem of momentum for one particle:,The derivative of the momentum of a particl
10、e with respect to time is equal to the force acting on the particle.This is the momentum theorem for one particle.,In a certain time interval,the change of the momentum of a particle is equal to the impulse of the force during the same interval of time.,Integral form.,The differential of the momentu
11、m of a particle equals the elementary impulse of the force acting on it.,Differential form:,12.3 Theorem of momentum,Projection form:,If,then is a constant vector and the particle is an inertial motion.,The law of conservation of the linear momentum of a particle,If,then is a const and the motion of
12、 the particle along the axis X is an inertial motion.,2.Theorem of momentum of a system of particles,the momentum theorem of a system of particles.,For the whole system of particles,we have,For any particle i in the system,we have.,The derivative of the linear momentum of a system of particles with
13、respect to time is equal to the geometric sum of all the external forces acting on the system.,The differential of the linear momentum of a system of particles is equal to the geometric sum of the elementary impulses of all the external forces acting on the system.,During a certain time interval,the
14、 change in the linear momentum of a system of particles is equal to the geometric sum of the impulses of all the external forces acting on the system during the same time interval.,2)Integral form:,1)Differential form:,3)Projection form:,Only external forces can change the total momentum of a system
15、 of particles,while internal forces are incapable of changing it.They only may cause that the momenta of the particles or parts of it are exchanged between the particles.,4)The conservation law of the linear momentum of a system of particles:,A big triangular column of mass M is placed on a smooth h
16、orizontal plane,on the slope of it lying a small triangular column of mass m.Determine the displacement of the big triangular column when the small column slides down the shape to the end.,Example,Solution:,1 Choose the system composed of these two bodies as the object to be investigated.,Analysis o
17、f force,Assume that the velocity of the big triangular block is the velocity of the small triangular block with respect to the big triangular block is.,Analysis of motion,2 From the conservation law of the linear momentum in horizontal direction(the system being at rest at the initial moment)we obta
18、in,A fluid flow through a bend,the fluid velocities at sections A and B being respect.Determine the dynamic force(additionally the dynamic reaction)exerted by the fluid on the bend.Assume the fluid is incompressible,the flow rate Q(m3/s)is constant and density of the the fluid is(kg/m3).,Example,3 F
19、rom the momentum theorem of a system of particles,1 Choose the fluid between the sections A and B as the system of particles under consideration.2 The analysis of the force is shown in the figure below.Assume that the fluid AB moves to the position ab during the time t.,Solution:,Static reaction,dyn
20、amic reaction,When calculating we often use the projection form,The force contrary to the force R is just the dynamic force exerted by the fluid to the bend.,To a system of particles,using the momentum theorem of a system of particles we obtain:,If the mass of the system does not change,The formulas
21、 above are the theorem of motion of the center of mass(or the differential equations of the motion of the center of mass).The product of the acceleration of the center of mass of a system and the mass of the whole system is equal to the geometric sum of all external forces acting on the system,the p
22、rincipal vector of the external force system.,12.4 Theorem of motion of the center of mass,1.Projection forms:,System of rigid bodies:Denoting by the No.i the body of mass mi and velocity vci we have,3.The theorem of motion of the center of mass is an equivalent expression to the momentum theorem of
23、 a system,being similar in form to the differential equations of motion for one particle.For a system of particles in arbitrary motion,the center of mass of the system moves as if it would be a particle of mass equal to the mass of the whole system to which all the external forces acting on the syst
24、em are applied.,4.The conservation law of the motion of the center of mass:,2)Knowing the external forces acting on the system,determine the law of motion of the center of mass.,5.The theorem of motion of the center of mass may solve two types of dynamical problems:,1)Knowing the motion of the cente
25、r of mass of a system,determine the external forces acting on the system(including the reactions of constraints).,A motor body is mounted on a horizontal foundation.The mass of the rotor is m2 and the mass of stator is m1.The shaft of the rotor passes through the center of mass O1 of stator,but ther
26、e is a distance e from the center of mass O2 of the rotor to O1 due to an inaccuracy in the production.Determine the reaction forces of the constraints exerted on the mount of the motor by the foundation when the rotor is rotating with the angular velocity w.,Example,Solution 1 choose the motor as t
27、he system of particles to be investigated and the analysis of forces is shown in the figure 2 Analysis of motion:The acceleration a1 at the center of mass of the stator O1 is zero.The acceleration a2 at the center of mass of the rotor O2 is ew2(directed to O1).,3 According to the theorem of motion o
28、f the center of mass we have,The dynamic reaction caused by the deviation from the center is a periodic function of time.,There is a floating hoisting boat of weight P1=200kN with the jib of weight P2=10kN and length l=8m,The weight of the lifted load P3=20kN.Assume that at the initial moment the wh
29、ole system at rest and the angle between the jib OA and the vertical was 1=60,Neglecting the resistance of the water determine the displacement of the boat at the instant when the jib OA makes an angle 2=30 to the vertical.,Example,1 Choose the system consisting of the hoisting boat with the jib and
30、 the load as the object under study.,2 The analysis of the force is shown in the figure.and because the whole system was initially at rest the coordinate of the center of mass of the whole system will remain at rest.,Solution:,The displacement of the boat is x.The displacement of the jib is,The displacement of the weight is,The negative sign shows that the boatmoves to the left.,The end,
