Kinematics of particles

Ch. 8: Kinetics of Particles
8.0 Outline  Introduction  Newton’s Second Law  Equations of Motion  Rectilinear Motion  Curvilinear Motion 415 416 417 418 421 444

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8.0 Outline

Ch. 8: Kinetics of Particles
8.1 Introduction Kinetics is the study of the relations between the forces and the motion. Here we will not seriously concern whether the forces cause the motion or the motiongenerates the forces (causality). In this chapter, the focus is on the particles. That is the body whose physical dimensions are so small compared with the radius of curvature of its path. There are at least 3 approaches toe the solution of kinetic problems: (a) Newton’s second law (b) work and energy method (c) impulse and momentum method.

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8.1 Introduction

Ch. 8: Kinetics ofParticles
8.2 Newton’s Second Law

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F = ma m = mass (resistance to rate of change of velocity) of the particle F = resultant force acting on the particle a = resulting acceleration measured in a nonaccelerating frame of reference

For most engineering problems on earth, the acceleration measured w.r.t. reference frame fixed to the earth’s surface may be treated as absolute. And Newton’s 2ndlaw of motion holds. Newton’s 2nd law breaks when the velocities of the order of the speed of light are involved  theory of relativity 8.2 Newton’s Second Law

Ch. 8: Kinetics of Particles
8.3 Equation of Motion and Solution of Problems

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∑ F = ma

--- equation of motion

scalar components decomposition according to a specified coordinate

Two problems of dynamics (1) specifiedkinematic conditions, find forces  straightforward application of Newton’s law as algebraic equations (2) specified forces, find motion  Difficulty depends on the form of force function (t, s, v, a), as the solutions are found by solving a system of differential equations. For simple functions, we can find closed form solutions of motion as in rectilinear motion (sec. 2.2). 8.3 Equation of Motionand Solution

Ch. 8: Kinetics of Particles

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Unconstrained motion Motion of the particle is determined by its initial motion and the forces from external sources. It is free of constraints and so has three degrees of freedom to specify the position. Three scalar equations of motion would have to be applied and integrated to obtain the motion. Constrained motion Motion of the particle ispartially or totally determined by restraining guides, other than its initial motion and the forces from external sources. Therefore, all forces, both applied and reactive, that act on the particle must be accounted for in Newton’s law. The number of d.o.f. and equations are reduced regarding to the type of constraints. 8.3 Equation of Motion and Solution

Ch. 8: Kinetics of Particles
Freebody diagram All forces acting on the particle needed to be accounted in the equations of motion. Free body diagram unveils every force that acts on the isolated particle. Only after the FBD has been completed should the equations of motion be written. The appropriate coordinate axes and directions should be indicated and consistently used throughout the problem.

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Treatment of the body asparticle is valid when the forces may be treated as concurrent through the mass center.

8.3 Equation of Motion and Solution

Ch. 8: Kinetics of Particles
8.4 Rectilinear Motion
If the x-axis is the direction of the rectilinear motion, = ma x= 0= 0 ∑ Fx ∑ Fy ∑ Fz If we are not free to choose a coordinate direction along the motion, the nonzero acceleration component will be shown up in allequations: = ma x ∑ Fx = ma y ∑ Fz ma z ∑ Fy = Other coordinate system such as n-t or r-θ a may be determined via the use of relative motion For pure translating moving reference frame a A = a B + a A/B

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8.4 Rectilinear Motion

Ch. 8: Kinetics of Particles
P. 8/1

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The coefficient of static friction between the flat bed of the truck and the crate it carries is 0.30. Determine...