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# Lect_17 - ACT Force field The pictures below show force...

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Lecture 17 Energy Diagrams ACT: Force field The pictures below show force vectors at different points in space for two different forces. Which of these forces is conservative? A. 1 B. 2 C. Both 1 2 W = 0 W 1 W = 0 W 3 W 2 W = 0 1 2 | W 2 + W 3 |= | W 1 | W loop = 0 | W 2 ’| > | W 1 ’| W loop 0 Work along a closed trajectories (loops). W = 0 W 2 W = 0 W 1 Relation between Relation between U and and F (1D) final initial x W F dx U = = −∆ For a conservative force (in 1D), constant x U F dx = − + x dU F dx = − 2 Examples: 1 2 y x dU U mgy F mg dy dU U kx F kx dx = = − = − = = − = − The force is minus the slope of the U ( x ) curve.

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Example: Spring 2 1 ( ) 2 U x kx = x U x x = 0, F = 0 dU/dx = 0 Potential energy of a box attached to a spring on a horizontal, frictionless table. x U x x < 0, F > 0 dU/dx < 0 x U x x > 0, F < 0 dU/dx > 0 x U x x > 0, F < 0 and larger in magnitude dU/dx > 0 and steeper than before The force always points “downhill”!!!
ACT: ACT: U vs F A Which of the force versus position graphs matches the potential energy function shown in blue? U x F x x x F F B C + Slope: 0 0 0 Force = slope Relation between Relation between U and and F (3D) (3D) x U F x = 1D: , , in cartesian coordinates for the radial component in spherical coordinates x y z r U U U F F F x y z U F r = − = − = − = − 3D: (minus the gradient of ) F U U = −∇ G G (“downhill”) (“downhill in the steepest direction”) Example Example

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Lect_17 - ACT Force field The pictures below show force...

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