1. ENGR 3340: Fundamentals of Statics and Dynamics Fundamentals of Statics and Dynamics - ENGR 3340 Professor: Dr. Omar E. Meza Castillo omeza@bayamon.inter.edu http://facultad.bayamon.inter.edu/omeza Department of Mechanical Engineering

2. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 2 Tentative Lectures Schedule TopicLecture Kinematics of a Particle 14

3. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Curvilinear Motion Topic 14: Kinematics of a Particle 3 One thing you learn in science is that there is no perfect answer, no perfect measure. A. O. Beckman

4. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 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. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 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. 5

6. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 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. General Curvilinear Motion. 6

7. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 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 General Curvilinear Motion. 7

8. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 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 = x i + y j + z k 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 8

9. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 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 9

10. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 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 10

11. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 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 11

12. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Motion of a Projectile Horizontal Motion: Since a x =0, Vertical Motion: Since a y =-g, 12

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24. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 24 ¿Preguntas? Comentarios

25. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 25 GRACIAS