14.1 D'Alembert'sprincipleofaparticleExample 1:The pendulum length of the simple pendulum is L,the mass of the pendulum is M. Calculate the differentialequation of the pendulum motion and the tension of the rope
Example 1: The pendulum length of the simple pendulum is L, the mass of the pendulum is M. Calculate the differential equation of the pendulum motion and the tension of the rope. 14.1 D'Alembert's principle of a particle
14.1 D'Alembert'sprincipleofaparticleAnswer:1. Force analysis and motion analysis: Theacceleration of the particleKaaa, =l0, a, =l2202. According to the acceleration analysis, addthe inertial force, and the magnitude is:F°=mlé, Fg =mle?3. By D'Alembert theorem to set up theFgnequilibrium equation:PZF,=0 F°+mgsin=0F8 +mgcos0-F =0ZF, =00+gDifferentialequations of motion for asin 0=01simplependulumFr = Pcos0+mli2Tension of the rope
2 , F ml F g ml n g = = FT P g Fn g Fτ an a 2. According to the acceleration analysis, add the inertial force, and the magnitude is: 3. By D'Alembert theorem to set up the equilibrium equation: 1. Force analysis and motion analysis; The acceleration of the particle 2 , a l a l = n = F = 0 F + mgsin = 0 g = 0 + cos − T = 0 g Fn Fn mg F Answer: + sin = 0 l g 2 cos F P ml T = + ——Differential equations of motion for a simple pendulum ——Tension of the rope 14.1 D'Alembert's principle of a particle
14.1 D'Alembert'sprincipleofaparticleExample 2: The cylinder of the ball mill rotates aroundthe horizontal axis O at the uniform angular velocity 0, FgFand the steel ball and materials to be crushed are installedinside. The steel ball is carried to a certain height by thecylinder wall to escape from the cylinder wall, and thenOmgfalls freely along the parabolic trajectory to smash thematerials. Set the radius of the inner wall of the cylinderas r in the figure, and try to find the escape Angle α ofthe steel ball.Answer: A steel ball that has not been detached from the cylinder walis taken as the research object, and the force is shown in the figure.Before the ball is detached from the cylinder wall, it moves ina circle, and its acceleration isa, =0a, =ro?The magnitude of the inertial force F isFg =mro
M F g F mg N r O Example 2: The cylinder of the ball mill rotates around the horizontal axis O at the uniform angular velocity , and the steel ball and materials to be crushed are installed inside. The steel ball is carried to a certain height by the cylinder wall to escape from the cylinder wall, and then falls freely along the parabolic trajectory to smash the materials. Set the radius of the inner wall of the cylinder as in the figure, and try to find the escape Angle of the steel ball. r Answer: A steel ball that has not been detached from the cylinder wall is taken as the research object, and the force is shown in the figure. Before the ball is detached from the cylinder wall, it moves in a circle, and its acceleration is a = 0 2 an = r The magnitude of the inertial force is g F 2 F mr g = 14.1 D'Alembert's principle of a particle
14.1 D'Alembert'sprincipleofaparticleSuppose to add the inertial forces, by D'Alembert'sprinciple, we have:ZF,=0N+mgcos0-Fg=0LroSo:N=mcosO)gThis is the normal reaction force suffered by the steel ballat any position . Obviously, when the steel ball is separatedfrom the cylinder wall, N = O, and thus, its escape angle αcan be obtained as:α=arccog
So： ( cos ) 2 = − g r N mg This is the normal reaction force suffered by the steel ball at any position . Obviously, when the steel ball is separated from the cylinder wall, , and thus, its escape angle can be obtained as: N = 0 arccos( ) 2 g r = Suppose to add the inertial forces, by D'Alembert's principle, we have: Fn = 0 + cos − = 0 g N mg F 14.1 D'Alembert's principle of a particle
14.2 D'Alembert's principle forparticle systemsMass m,,AcceleratedvelocityaNparticlesParticlelforma systemExternal force F, internalforceF,,F=-ma,of particles:Whatkind ofD'Alembert'sFe + Fi + F8 =0force systemprinciple ofaparticle:is formed?Whatkind ofZF ++ZF8 =0force systemNparticlesis formed?constitutetheparticle system:Zmo(Fe)+Em.(F)+Zmo(F°)=0ZFe +ZF8 =0Zmo(Fe)+Emo(F°)= 0
N particles form a system of particles: Particle i 0 e i g F F F i i i + + = 14.2 D'Alembert's principle for particle systems + + = 0 g i i i e Fi F F External force , internal force , e Fi i Fi Mass mi , Accelerated velocity i a i i g Fi m a = − D'Alembert's principle of a particle： N particles constitute the particle system: What kind of force system is formed？ ( ) + ( ) + ( ) = 0 g O i i O i e mO Fi m F m F ( ) ( ) 0 e g + = m F m F O i O i0 e g + = F F i i What kind of force system is formed？