chapt5 - 3. Kinetics of Particles: Energy and Momentum...

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3. Kinetics of Particles: Energy and Momentum Methods
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Introduction • Previously, problems dealing with the motion of particles were solved through the fundamental equation of motion, Current chapter introduces two additional methods of analysis. . a m F G G = Method of work and energy : directly relates force, mass, velocity and displacement. Method of impulse and momentum : directly relates force, mass, velocity, and time.
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Work of a Force • Differential vector is the particle displacement . r d G Work of the force is dz F dy F dx F ds F r d F dU z y x + + = = = α cos G G • Work is a scalar quantity, i.e., it has magnitude and sign but not direction. force. length × • Dimensions of work are Units are ( ) ( )( ) m 1 N 1 J 1 = joule
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Work of a Force • Work of a force during a finite displacement, () ( ) + + = = = = 2 1 2 1 2 1 2 1 cos 2 1 A A z y x s s t s s A A dz F dy F dx F ds F ds F r d F U α G G • Work is represented by the area under the curve of F t plotted against s .
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Work of a Force • Work of a constant force in rectilinear motion, ( ) x F U Δ = α cos 2 1 • Work of the force of gravity, () y W y y W dy W U dy W dz F dy F dx F dU y y z y x Δ = = = = + + = 1 2 2 1 2 1 • Work of the weight is equal to product of weight W and vertical displacement Δ y. • Work of the weight is positive when Δ y < 0, i.e., when the weight moves down.
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Work of a Force • Magnitude of the force exerted by a spring is proportional to deflection, () lb/in. or N/m constant spring = = k kx F • Work of the force exerted by spring , 2 2 2 1 2 1 2 1 2 1 2 1 kx kx dx kx U dx kx dx F dU x x = = = = • Work of the force exerted by spring is positive when x 2 < x 1 , i.e., when the spring is returning to its undeformed position. • Work of the force exerted by the spring is equal to negative of area under curve of F plotted against x , () x F F U Δ + = 2 1 2 1 2 1
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Work of a Force Work of a gravitational force (assume particle M occupies fixed position O while particle m follows path shown), 1 2 2 2 1 2 2 1 r Mm G r Mm G dr r Mm G U dr r Mm G Fdr dU r r = = = =
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Work of a Force Forces which do not do work (ds = 0 or cos α = 0) : • weight of a body when its center of gravity moves horizontally. • reaction at a roller moving along its track, and • reaction at frictionless surface when body in contact moves along surface, • reaction at frictionless pin supporting rotating body,
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dv mv ds F ds dv mv dt ds ds dv m dt dv m ma F t t t = = = = = • Consider a particle of mass m acted upon by force F G • Integrating from A 1 to A 2 , energy kinetic mv T T T U mv mv dv v m ds F v v s s t = = = = = 2 2 1 1 2 2 1 2 1 2 1 2 2 2 1 2 1 2 1 The work of the force is equal to the change in kinetic energy of the particle . F
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chapt5 - 3. Kinetics of Particles: Energy and Momentum...

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