Lecture14-2A

Lecture14-2A - Today's Lecture Lecture 14 Chapter 8 Energy...

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Today Today s Lecture s Lecture Lecture 14: Lecture 14: Chapter 8, Chapter 8, Energy Diagrams Energy Diagrams Conservation of Energy Conservation of Energy Chapter 10, Chapter 10, System of Particles System of Particles
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Example: Force and Potential Energy Example: Force and Potential Energy (a) Derive an expression for the potential energy of an object subject to a force, F = ax- bx 3 , where a = 5 N/m and b = 2 N/m 3 assuming that U(0) = 0. (b) Derive the turning points when E = -1J . U x U 0 0 x F x dx 0 x ax bx 3 dx U x 1 2 ax 2 1 4 bx 4 First we note that the force vanishes at: x 0 and x  a / b This potential is double welled with a local maximum at x = 0 . The minimum value of the potential is: U a / b 1 2 a a b 1 4 b a 2 b 2 1 4 a 2 b U 5/2 25 8 3.125 J The potential energy is found from the integral:
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Example: Force and Potential Energy Example: Force and Potential Energy Derive an expression for the potential energy of a particle subject to a force, F = ax-bx 3 , where a = 5 N/m and b = 2 N/m 3 assuming that U(0) = 0. Derive the turning points when E = -1J . Since 0 > E > U min the particle is bound in one of the two wells with a velocity given by: The maximum velocity occurs when U = U min . The turning points occur when v = 0 or E = U . Solving the quartic yields: U x 1 2 ax 2 1 4 bx 4 E K U 1 2 mv 2 U v 2 E U / m 1 5 2 x 2 1 2 x 4 x  .662, 2.14
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Example: Force and Potential Energy Example: Force and Potential Energy Derive an expression for the potential energy of a particle subject to a force, F = ax-bx 3 , where a = 5 N/m and b = 2 N/m 3 assuming that U(0) = 0. Derive the turning points when E = -1J . U x 1 2 ax 2 1 4 bx 4 The turning points for x > 0 are the ones shown in the figure. What about the turning points for x < 0 ? The particle oscillates between these two points! What if E > 0 ? The turning points are: x  .662, 2.14
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Conservation of Energy From the Work-Energy Theorem the work done on an object is equal to the change in its kinetic energy: W m d v dt d r F net d r Δ K 1 2 mv f 2 1 2 mv i 2 If we consider separately the work done by conservative forces, W c , and non-conservative forces, W nc : Δ K W c W nc Δ K Δ U W nc Potential energy was defined as the negative of the work done by conservative forces: D U = - W c . Hence: In the absence of non In the absence of non -conservative forces, the total mechanical energy is conservative forces, the total mechanical energy is conserved! conserved!
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Lecture14-2A - Today's Lecture Lecture 14 Chapter 8 Energy...

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