1. Lecture 2 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 2 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 2 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 2 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 2 MECN 4600 Inter - Bayamon 12.4 General Curvilinear Motion. Curvilinear motion occurs when a particle moves along a curved path. a. Position: The position of the particle, measured from a fixed point O, will be designated by the position vector r=r(t). The magnitude and the direction change as the particle moves along the curve. b. Displacement: The displacement Δr represent the change in the particle’s position and is determined by vector subtraction.

6. Lecture 2 MECN 4600 Inter - Bayamon c. Velocity: During the time Δt, the average velocity of the particle during this time interval is The instantaneous velocity is determined from this equation by letting Δt -> 0, an consequently the direction of Δr approaches the tangent to the curve. Hence, The velocity can be positive (+) or negative (-). The magnitude of the velocity is called speed, and it is generally expressed in units of m/s or ft/s. 12.4 General Curvilinear Motion.

7. Lecture 2 MECN 4600 Inter - Bayamon d. Acceleration: If the particle has a velocity of v at time t and a velocity v’=v+Δv at t+Δt, then the average acceleration of the particle during the time interval Δt, is defined as The Δv = v’ - v represents the difference in the velocity during the time interval Δt The instantaneous acceleration is a vector defined as 12.4 General Curvilinear Motion.

8. Lecture 2 MECN 4600 Inter - Bayamon 12.5 General Curvilinear Motion : Rectangular Components a. Position: If the particle is at point (x,y,z) on the curved path s shown in figure, then its location is defined by the position vector. r = xi + yj + zk When the particles moves, the x,y,z components of r will be functions of time; i.e., x=x(t), y=y(t), z=z(t), so that r=r(t). And the direction of r is specified by the unit vector u r =r/r The magnitude of r is defined by

9. Lecture 2 MECN 4600 Inter - Bayamon 12.5 General Curvilinear Motion : Rectangular Components b. Velocity: The first time derivative of r yields the velocity of the particle. Hence The “dot” notation represent the first time derivatives of x=x(t), y=y(t), z=z(t), respectively. The magnitude of v is defined by and a direction that is specified by the unit vector

10. Lecture 2 MECN 4600 Inter - Bayamon 12.5 General Curvilinear Motion : Rectangular Components c. Acceleration: The acceleration of the particle is obtained by taking the first time derivative of v (or the second time derivative of r). We have Where, The magnitude of a is defined by and a direction that is specified by the unit vector

11. Lecture 2 MECN 4600 Inter - Bayamon 12.6 Motion of a Projectile The free-flight motion of a projectile is often studied in terms of its rectangular components. To illustrate the kinematic analysis, consider a projectile launched at point (x 0,y 0 ), with a initial velocity of v0, having components (v 0 ) x and (v 0 ) y. The air resistance is neglected and the only force acting on the projectile is its weight, which causes the projectile to have a constant downward acceleration of approximately a c =g = 9.81m/s 2 = 32.2ft/s 2

12. Lecture 2 MECN 4600 Inter - Bayamon 12.6 Motion of a Projectile Horizontal Motion: Since a x =0, Vertical Motion: Since a y =-g,

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24. Lecture 2 MECN 4600 Inter - Bayamon Due, Tuesday, February 06, 2012 Omar E. Meza Castillo Ph.D. Homework2  WebPage 24