Chapter_14_mod - Chapter14:KineticsofaParticle:...

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1 Chapter 14: Kinetics of a Particle: Work and Energy Chapter 14 Objectives To develop the principle of work and energy and apply it to solve problems that involve force, velocity and displacement To study problems that involve power and efficiency To introduce the concept of a conservative force and apply the theorem of conservation of energy to solve kinetic problems
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2 14.1: The Work of a Force This chapter focuses on the work energy principle This is very useful for solving problems that involve forces, velocities and displacement Work definition: a force F will do work on a particle only when the particle undergoes a displacement in the direction of the force Note that the particle can move in a component direction of the force Work Illustration For instance, a force F move the particle from r to r’ The displacement is dr with a magnitude of ds The work is then defined by: ݀ݑ ൌ ܨ݀ݏܿ݋ݏߠ This is a scalar quantity Note that dr is a vector
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3 Dot Product Mathematically, the dot product can be used to calculate the work on a particle, as ݀ݑ ൌ ࡲ ∙ ݀࢘ The dot product produces a scalar and only multiplies components that are in the same direction (i.e. ࢏∙࢏ൌ1 ,࢏∙࢐ൌ0 ) If the angle between the force and displacement is between 0 0 and 90 0 , the work of that force is positive If the angle is between 90 0 and 180 0 then the work of that force is negative If the angle is 90 0 the work of that force is zero Work of a Variable Force If the particle that is acted upon by a force undergoes a finite displacement from r 1 to r 2 then the work of the force can be quantified by ܷ ଵ→ଶ ൌනࡲ∙݀࢘ ൌනܨܿ݋ݏߠ݀ݏ This formulation can only be used if the F and theta are known as a function of position
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4 Work of a Constant Force Moving Along a Straight Line If the force has a constant magnitude and direction from the straight lint path of displacement then the work can be defined as ܷ ଵ→ଶ ൌܨ ܿ݋ݏߠ න ݀ݏ ܿ݋ݏߠሺݏ െݏ Work of a Weight The work of a weight only has an effect if a displacement occurs in the y direction (the direction aligned with gravity) ܷ ଵ→ଶ ൌെܹ ݕ െݕ ൌെܹΔݕ A particle that moves upward has negative work, a particle that moves downward has positive work Note that the particle may move in the x/z directions
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5 Work of a Spring Force The work of a spring acting on a particle can be calculated from the force required to displace the spring ܷ ଵ→ଶ ൌනܨ ݀ݏ ൌെ 1 2 ݇ݏ 1 2 The signs of this quantity can be confusing If starting from the un stretched length, no matter what direction the displacement is in, the work is negative If starting from a stretched length the work can be positive or negative (does the spring get further stretched or less
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Chapter_14_mod - Chapter14:KineticsofaParticle:...

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