Lecture 4 Vector Mechanics for Engineers: Dynamics MECN 3010 Department of Mechanical Engineering Inter American University of Puerto Rico Bayamon Campus

LectureLecture

44Vector Mechanics for Engineers: Vector Mechanics for Engineers:

DynamicsDynamics MECN 3010 MECN 3010

Department of Mechanical EngineeringDepartment of Mechanical Engineering

Inter American University of Puerto RicoInter American University of Puerto Rico

Bayamon CampusBayamon Campus

Dr. Omar E. Meza CastilloDr. Omar E. Meza [email protected]

http://www.bc.inter.edu/facultad/omeza

Lecture 4Lecture 4MEC

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Tentative Lecture ScheduleTentative Lecture Schedule

TopicTopic LectureLecture

Kinematics of a ParticleKinematics of a Particle 1,2,3,41,2,3,4

Kinetics of a Particle: Force and AccelerationKinetics of a Particle: Force and Acceleration

Kinetics of a Particle: Work and EnergyKinetics of a Particle: Work and Energy

Kinetics of a Particle: Impulse and MomentumKinetics of a Particle: Impulse and Momentum

Planar Kinematics of a Rigid BodyPlanar Kinematics of a Rigid Body


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Force and AccelerationForce and Acceleration

Topic 1: Kinematics of a Topic 1: Kinematics of a ParticleParticle

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"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 AlbertEinstein Albert


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Chapter ObjectivesChapter Objectives

To state Newton’s Second Law of Motion To state Newton’s Second Law of Motion and to define mass and weight.and to define mass and weight.

To analyze the accelerated motion of a To analyze the accelerated motion of a particle using the equation of motion with particle using the equation of motion with different coordinate system.different coordinate system.

To write the equation of motion for an To write the equation of motion for an accelerating body.accelerating body.

To draw the free-body and kinetic To draw the free-body and kinetic diagrams for an accelerating body.diagrams for an accelerating body.

To investigate central-force motion and To investigate central-force motion and apply it to problems in space mechanics.apply it to problems in space mechanics.

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13.1 Newton’s Law of Motion13.1 Newton’s Law of Motion

APLICATIONSAPLICATIONS The motion of an The motion of an

object depends on the object depends on the forces acting on it.forces acting on it.

A parachutist relies A parachutist relies on the atmospheric on the atmospheric drag resistance force drag resistance force to limit his velocity.to limit his velocity.

Knowing the drag Knowing the drag force, how can we force, how can we determine the determine the acceleration or acceleration or velocity of the velocity of the parachutist at any parachutist at any point in time?point in time?

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13.1 Newton’s Law of Motion13.1 Newton’s Law of Motion

NEWTON’S LAWS OF MOTIONNEWTON’S LAWS OF MOTIONThe motion of a particle is governed by The motion of a particle is governed by Newton’s three laws of motion.Newton’s three laws of motion.

First Law: First Law: A particle originally at rest, or A particle originally at rest, or moving in a straight line at constant velocity, moving in a straight line at constant velocity, will remain in this state if the resultant force will remain in this state if the resultant force acting on the particle is zero.acting on the particle is zero.

Second Law: Second Law: If the resultant force on the If the resultant force on the particle is not zero, the particle experiences an particle is not zero, the particle experiences an acceleration in the same direction as the acceleration in the same direction as the resultant force. This acceleration has a resultant force. This acceleration has a magnitude proportional to the resultant force.magnitude proportional to the resultant force.

Third Law: Third Law: Mutual forces of action and reaction Mutual forces of action and reaction between two particles are equal, opposite, and between two particles are equal, opposite, and collinear.collinear.

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13.2 Newton’s Second Law of Motion13.2 Newton’s Second Law of Motion

The first and third laws were used in developing The first and third laws were used in developing the concepts of statics. Newton’s second law the concepts of statics. Newton’s second law forms the basis of the study of dynamics.forms the basis of the study of dynamics.

Mathematically, Mathematically, Newton’s second law of motion Newton’s second law of motion can be written:can be written:

where where FF is the is the resultant unbalanced force resultant unbalanced force acting acting on the particle, and on the particle, and aa is the is the accelerationacceleration of the of the particle. The positive scalar m is called the particle. The positive scalar m is called the massmass of the particle.of the particle.

Newton’s second law cannot be used when the Newton’s second law cannot be used when the particle’s speed approaches the speed of light, or particle’s speed approaches the speed of light, or if the size of the particle is extremely small (~ size if the size of the particle is extremely small (~ size of an atom).of an atom).

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13.3 Newton’s Law of Gravitational Attraction13.3 Newton’s Law of Gravitational Attraction

Any two particles or bodies have a Any two particles or bodies have a mutuallymutually attractive gravitational force acting between attractive gravitational force acting between them. Newton postulated the law governing this them. Newton postulated the law governing this gravitational force as:gravitational force as:

When near the surface of the earth, the only When near the surface of the earth, the only gravitational force having any sizable magnitude gravitational force having any sizable magnitude is that between the earth and the body. This force is that between the earth and the body. This force is called the weight of the body.is called the weight of the body.

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13.4 Mass and Weight13.4 Mass and Weight

It is important to understand the difference It is important to understand the difference between the mass and weight of a body!between the mass and weight of a body!

MassMass is an is an absolute property absolute property of a body. It is of a body. It is independent of the gravitational field in which it is independent of the gravitational field in which it is measured. The mass provides a measure of the measured. The mass provides a measure of the resistance of a body to a change in velocityresistance of a body to a change in velocity, as , as defined by Newton’s second law of motion (m = defined by Newton’s second law of motion (m = FF//aa).).

The weight of a body is The weight of a body is not absolutenot absolute, since it , since it depends on the gravitational field in which it is depends on the gravitational field in which it is measured. measured. WeightWeight is defined as is defined as

where g is the where g is the acceleration due to gravityacceleration due to gravity..

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13.5 Units: SI System vs. FPS System13.5 Units: SI System vs. FPS System

SI system: SI system: In the SI system of units, In the SI system of units, massmass is a is a base unitbase unit and and weightweight is a is a derived unitderived unit. Typically, . Typically, mass is specified in kilograms(kg), and weight is mass is specified in kilograms(kg), and weight is calculated from W = mg. If the gravitational calculated from W = mg. If the gravitational acceleration (g) is specified in units of m/sacceleration (g) is specified in units of m/s22, then , then the weight is expressed in the weight is expressed in newtonsnewtons (N). On the (N). On the earth’s surface, g can be taken as g = 9.81 m/searth’s surface, g can be taken as g = 9.81 m/s22..

W (N) = m (kg) g (m/sW (N) = m (kg) g (m/s22) => N = kg·m/s) => N = kg·m/s22

FPS System: FPS System: In the FPS system of units, In the FPS system of units, weightweight is is a a base unit base unit and and massmass is a is a derived unitderived unit. Weight is . Weight is typically specified in typically specified in pounds pounds (lb), and mass is (lb), and mass is calculated from m=W/g. If g is specified in units of calculated from m=W/g. If g is specified in units of ft/sft/s22, then the mass is expressed in , then the mass is expressed in slugsslugs. On the . On the earth’s surface, g is approximately 32.2 ft/searth’s surface, g is approximately 32.2 ft/s 2 2..

m (slugs) = W (lb)/g (ft/sm (slugs) = W (lb)/g (ft/s 2 2) => slug = lb·s) => slug = lb·s 2 2 /ft/ft

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13.6 The Equation of Motion13.6 The Equation of Motion

The motion of a particle is governed by Newton’s The motion of a particle is governed by Newton’s second law, relating the unbalanced forces on a particle second law, relating the unbalanced forces on a particle to its acceleration. If more than one force acts on the to its acceleration. If more than one force acts on the particle, the equation of motion can be writtenparticle, the equation of motion can be written

where where FFRR is the resultant force, which is a vector is the resultant force, which is a vector summation of all the forces.summation of all the forces.

To illustrate the equation, consider a particle acted on To illustrate the equation, consider a particle acted on by two forces Fby two forces F11 and F and F22..

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13.7 Equation of Motion for a System of Particles13.7 Equation of Motion for a System of Particles

The equation of motion will now be extended to include a system of particles isolated within an enclosed region in space, as shown in figure.

As in statics, there are internal forces fi and external forces Fi.

∑Fi= ∑miai

Fifi=

miai

Free-bodydiagram

Kineticdiagram

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13.8 Equation of Motion: 13.8 Equation of Motion: Rectangular CoordinatesRectangular Coordinates

When a particle moves relative to an inertial x,y,z frame of reference, the forces acting on the particle, as well as its acceleration, can be expressed in terms of their i,j,k components.

Consequently, we may write the following three scalar equations:

∑Fx= max

∑Fy= may

∑Fz= maz

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13.9 Procedure for the Application of the Equation of Motion13.9 Procedure for the Application of the Equation of Motion

Select a convenient Select a convenient inertial coordinate systeminertial coordinate system. . Rectangular, normal/tangential, or cylindrical Rectangular, normal/tangential, or cylindrical coordinates may be used.coordinates may be used.

Draw a Draw a free-body diagram free-body diagram showing all showing all external external forces forces applied to the particle. Resolve forces into applied to the particle. Resolve forces into their appropriate components.their appropriate components.

Draw the Draw the kinetic diagramkinetic diagram, showing the particle’s , showing the particle’s inertial force, minertial force, maa. Resolve this vector into its . Resolve this vector into its appropriate components.appropriate components.

Apply the Apply the equations of motion equations of motion in their scalar in their scalar component form and solve these equations for the component form and solve these equations for the unknowns.unknowns.

It may be necessary to apply the proper It may be necessary to apply the proper kinematic kinematic relationsrelations to generate additional equations. to generate additional equations.

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Theory: Absolute Dependent Motion Analysis of Two ParticlesTheory: Absolute Dependent Motion Analysis of Two Particles

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Omar E. Meza Castillo Ph.D.Omar E. Meza Castillo Ph.D.

Homework3 Homework3 WebPage WebPage

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Lecture 4 Vector Mechanics for Engineers: Dynamics MECN 3010 Department of Mechanical Engineering Inter American University of Puerto Rico Bayamon Campus