Chapter_15_mod - Chapter15:KineticsofaParticle:...

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1 Chapter 15: Kinetics of a Particle: Impulse and Momentum Chapter 15 Objectives To develop the principle of linear impulse and momentum for a particle and apply it to solve problems that involve force, velocity and time To study the conservation of linear momentum for particles To analyze the mechanics of impact To introduce the concepts of angular impulse and momentum
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2 15.1: Principles of Linear Impulse and Momentum This chapter focuses on the impulse momentum principle This is very useful for solving problems that involve forces, velocities and times Equation of motion for a particle is: ෍ࡲൌ݉ࢇൌ݉ ݀࢜ ݀ݐ If this equation is integrated with respect to time we get: ෍නࡲ݀ݐ ൌ݉න݀࢜ ൌ݉ݒ െ݉ݒ This is the principle of linear impulse and momentum Linear Momentum Linear momentum is defined as ࡸൌ݉࢜ This is a vector with the same direction as the velocity, and has units of kg*m/s or slug*ft/s Its magnitude is equal to the mass multiplied by the velocity Note that the linear impulse and momentum equation can only be applied if the forces that are acting on the particle are constant with time or can be expressed as a function of time
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3 Linear Impulse The linear impulse is defined as: ࡵൌ ׬ ࡲ݀ݐ This is a vector that has the same direction as the net force and has units of N*s or lb*s The magnitude of this is force multiplied by the time that it acts over Note that the units of impulse and momentum are the same, although by convention they are written differently If the force is constant with time, the linear impulse becomes: ݀ݐ ൌࡲ ݐ െݐ Graphical Representation of Linear Impulse
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4 Principle of Linear Impulse and Momentum For most problems, it is convenient to re write the equations in vector form ݉ ݒ ௫ଵ ൅ ෍නܨ ݀ݐ ൌ݉ ݒ ௫ଶ ݉ ݒ ݀ݐ ݒ ݉ ݒ ௭ଵ ݀ݐ ݒ ௭ଶ Procedure for Analysis Free Body Diagram Establish the inertial reference frame Establish the particles initial and final velocity Principle of Impulse and Momentum Break to vectors into components Solve for the unknowns
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5 15.2: Principle of Linear Impulse and Momentum for a System of Particles If a system of particles can be related, then the equation of motion for the system is ෍ ࡲ ൌ ෍ ݉ ݀࢜ ݀ݐ The internal forces will cancel, so we only sum the external forces The summation on the right hand side, locates the mass center for us and therefore we can re write the impulse/momentum equation as ݉ ݒ ீଵ ൅ ෍නܨ ݀ݐ ൌ݉ ݒ ீଶ Problem 15 5 If cylinder A is given an initial downward speed of 2 m/s, determine the speed of each cylinder when ݐൌ3ݏ . Neglect the mass of the pulleys.
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6 Solution to 15 5 Solution to 15 5 cont.
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Chapter_15_mod - Chapter15:KineticsofaParticle:...

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