1. Lecture 3 Vector Mechanics for Engineers: Dynamics MECN 3010 Department of Mechanical Engineering Inter American University of Puerto Rico Bayamon Campus Dr. Omar E. Meza Castillo omeza@bayamon.inter.edu http://www.bc.inter.edu/facultad/omeza

2. Lecture 3 MECN 4600 Inter - Bayamon 2 Tentative Lecture Schedule TopicLecture Kinematics of a Particle 1 Kinetics of a Particle: Force and Acceleration Kinetics of a Particle: Work and Energy Kinetics of a Particle: Impulse and Momentum Planar Kinematics of a Rigid Body

3. Lecture 3 MECN 4600 Inter - Bayamon Introduction and Basic Concepts Topic 1: Kinematics of a Particle 3 "Lo peor es educar por métodos basados en el temor, la fuerza, la autoridad, porque se destruye la sinceridad y la confianza, y sólo se consigue una falsa sumisión” Einstein Albert

4. Lecture 3 MECN 4600 Inter - Bayamon Chapter Objectives  To introduce the concepts of position, displacement, velocity, and acceleration.  To study particle motion along a straight line and represent this motion graphically.  To investigate particle motion along a curve path using different coordinate systems.  To present an analysis of dependent motion of two particles.  To examine the principles of relative motion of two particles using translating axes. 4

5. Lecture 3 MECN 4600 Inter - Bayamon 12.7 Curvilinear Motion : Normal and Tangential Components When the path along which a particle travels is known, then it is often convenient to describe the motion using n and t coordinate axes which act normal and tangent to the path a. Planar Motion: The t axis is tangent to the curvature, and n is perpendicular to the t axis. ρ(rho) is the radius of curvature and O’ is the center of curvature. The plane which contains the n and t axes is referred as the embracing or osculating plane. Radius of curvature ds

6. Lecture 3 MECN 4600 Inter - Bayamon b. Velocity: Since the particle moves, s is a function of time. The particle’s velocity v has a direction that is always tangent to the path, and a magnitude that is determined by taking the time derivative of the path function s=s(t), i.e., v= ds/dt. Hence v=vu t where 12.7 Curvilinear Motion : Normal and Tangential Components

7. Lecture 3 MECN 4600 Inter - Bayamon c. Acceleration: The acceleration of the particle is the time rate of change of the velocity. Thus wherewhere 12.7 Curvilinear Motion: Normal and Tangential Components

8. Lecture 3 MECN 4600 Inter - Bayamon 12.8 Curvilinear Motion: Cylindrical Components a. Polar Coordinates: If motion is restricted to the plane, then polar coordinates is used. We can specify the location of the particle shown in figure using a radial coordinate r(u r ), which extend outward from the fixed origin O to the particle, and a transverse coordinate θ(u θ ), which is the counterclockwise angle between a fixed reference line and the r axis. b. Position: At any instant the position of the particle, is defined by the position vector r = ru r

9. Lecture 3 MECN 4600 Inter - Bayamon 12.8 Curvilinear Motion: Cylindrical Components c. Velocity: The instantaneous velocity v is obtained by taking the time derivative of r. Using a dot to represent the time derivative, we have wherewhere

10. Lecture 3 MECN 4600 Inter - Bayamon 12.8 Curvilinear Motion: Cylindrical Components d. Acceleration: Taking the time derivatives of the last velocity equations, we obtain the particle’s instantaneous acceleration wherewhere

11. Lecture 3 MECN 4600 Inter - Bayamon 12.8 Curvilinear Motion: Cylindrical Components e. Cylindrical Coordinates: IF the particles moves along a space curve as shown in figure, then its location may be specified by the three cylindrical coordinates r,θ,z. The position, velocity, and acceleration of the particle can be written in terms of its cylindrical coordinates as follows:

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27. Lecture 3 MECN 4600 Inter - Bayamon Due, Tuesday, February 06, 2012 Omar E. Meza Castillo Ph.D. Homework2  WebPage 27