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lecture 13

# lecture 13 - and determine if they are stable/unstable...

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1 Lecture 13 Relationship between conservative forces and PE Energy diagrams

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2 r F r F r F d dU dU d dW d W - = - = = = How are conservative forces related to potential energy? ( 29 dx x dU F x - = In 1-D: In 3-D: U -∇ = F
3 The elastic potential energy of a spring: ( 29 kx kx dx d dx x dU F x - = - = - = 2 2 1 (Hooke’s law) x (m) U (J) x 2 x 1 E

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4 Consider a particle of mass m with energy E as show in the graph. The object has “turning points” where E = U, that is when its kinetic energy is zero. These points occur at k E x kx U K E 2 2 1 0 2 ± = + = + = (classical turning points)
5 The equilibrium points (F net = 0) are found by determining the roots of ( 29 0 = dx x dU The equilibrium points can be stable or unstable ( 29 0 2 2 < = eq x x dx x U d ( 29 0 2 2 = eq x x dx x U d Unstable Equilibrium Stable Equilibrium

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6 Example: Consider the potential energy function U(x) = 26x 4 – 3x 2 . Determine the location of the equilibrium points

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Unformatted text preview: and determine if they are stable/unstable equilibrium points. ( 29 6 104 3 =-= x x dx x dU The roots of the cubic equation are x = 0, ± 0.24 m The equilibrium points are found by setting the first derivative of U(x) = 0: 7 The 2 nd derivative of the potential energy is ( 29 6 312 2 2 2-= x dx x U d At x = 0 the second derivative is less than 0 and at ± 0.24 m it is > 0. x = 0 is a point of unstable equilibrium x = ± 0.24 m are points of stable equilibrium Example continued 8-0.1-0.05 0.05 0.1 0.15 0.2-0.41-0.21-0.01 0.19 0.39 x (m) U (J) A plot of U(x) 9 Summary ( 29 dx x dU F x-= The work done by a conservative force may be expressed as a change in potential energy. r F d dU ⋅-=...
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lecture 13 - and determine if they are stable/unstable...

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