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lecturenotes-Feb15

lecturenotes-Feb15 - From Potential Energy Curves to Force...

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From Potential Energy Curves to Force Curves
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Finding a conservative force from potential energy F ( x ) U ( x ) In 1-D: Δ U ( x ) = W cons = Fdx x 1 x 2 where F is a slowly varying, internal force acting on a particle in system Δ U ( x ) ≈ − F ( x ) Δ x then Now go backwards, say you know the change in potential energy at some point and you want to know the force at that point… (in the differential limit) F ( x ) = dU ( x ) dx F grav ( y ) = dU grav ( y ) dy = mgy ( ) dy = mg F spring ( x ) = dU spring ( x ) dx = 1 2 kx 2 ( ) dx = kx
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Problem: Look at the figure for potential vs x. The force acts on a 0.50 kg particles and U A =3 J, U B =7 J, U C =9 J. (a) Calculate the magnitude and direction of the force in all five regions. (b) Draw the plot. 9 J 7 J 3 J 2N -6N What happens at x=2, x=4, x=5, and x=6? F(x)
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Potential energy (if conservative force) : Δ U grav = mg Δ y If U grav ( y = 0) 0 then U grav ( y ) = mgy Δ U spring = 1 2 kx f 2 1 2 kx i 2 If U spring ( x = 0) 0 then U spring ( x ) = 1 2 kx 2 W = −Δ U Quick Review Non-Conservative Force: W loop 0 Conservative Force: W loop = 0 Mechanical energy: Conservative System: E mech = KE + U E mech i = E mech f
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How to Apply Your Knowledge to Solve Problems? When Friction is present, it always removes energy, so E(final) is less the E(initial) E mec ( final ) = E mec ( initial ) Energy removed by work of friction K f + U f = K i + U i F f d Always think about the problem and the signs!! My preference is to always use Mechanical Energy E mec ( final ) = E mec ( initial ) K f + U f = K i + U i If there is no non-conservative force involved, use E mech
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Non-Conservative Forces: Friction Mechanical energy: Conservative System Friction takes Energy out of the system—treat it like thermal energy E mech = KE + U Δ E mech 0 Lose Mechanical energy In General the work done by an external force is W= Δ E= Δ E mech + Δ E Thermal If there is no work done by an external force Δ E mech + Δ E Thermal = 0 E mech ( final ) = E mech ( initial ) E th ( final ) + E th ( initial ) K f + U f = K i + U i F f displacement
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What happens if non-isolated or non-conservative forces? Friction? Motor?
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Friction: Loss Mechanical Energy Question #8: A block of mass m slides down the inclined plane starting with zero velocity. Region D has friction and it comes to rest after moving a distance D. Mechanical Energy is not Conserved. E mec ( final ) = E mec ( initial ) − Δ E th E mec ( final ) = E mec ( initial ) F f x We can draw this. E mec ( final ) = E mec ( initial ) − Δ E th E mec ( final ) = E mec ( initial ) F f x K = mv 2 2
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A new way to look at Friction Question #9: Find K and E mec as the block moves along. E mech -F f distance K = mv 2 2
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