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A potential energy function is given by U = A x3 + B x2 − C x + D . For positive parameters A, B, C, D this potential has two equilibrium positions, one stable and one unstable. Find the stable equilibrium position for A = 1.45 J/m3 , B = 2.85 J/m2 , C = 1.7 J/m , and D = 0.6 J . Answer in units of m

Respuesta :

cjmejiab

Answer:

7.8655

Step-by-step explanation:

Given the functionl

[tex]U=Ax^3+Bx^2-cx+D[/tex]

Where,

[tex]A=1.45J/m^3\\B=2.85J/m^2\\c= 1.7J/m\\D=0.6J[/tex]

We have, replacing,

[tex]U=1.45x^3+2.85x^2-1.7x+0.6[/tex]

We need to derivate and verify for stable equilibrum that,

[tex]\frac{d^2 U}{dx^2}>0[/tex]

Then,

[tex]U'(x) = 4.32x^2+5.7x-1.7[/tex]

Roots in,

[tex]x_1=-1.57008\\x_2=0.250636[/tex]

We obtain U"(x), then we have

[tex]\frac{d^2U}{dx}=8.64x+5.7[/tex]

For [tex]x=-1.57, U"(x)=-7.86549<0[/tex] Unestable

For [tex]x=0.250636 U"(x)=7.8655>0[/tex] Stable

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