Question

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 = 3.65 J/m3 , B = 3.25 J/m2 , C = 2.1 J/m , and D = 3.6 J . Answer in units of m.

Answer 1

Given Information:

Function = U = Ax³ + Bx² − Cx + D

A = 3.65 J/m³

B = 3.25 J/m²

C = 2.1 J/m

D = 3.6 J

Required Information:

Stable equilibrium position = X = ?

Answer:

Stable equilibrium position = X₂ = 0.232

Step-by-step explanation:

Stability can be determined by analyzing the slope of the function. First we will find out the first derivative of the given function and the equilibrium position will be at dU/dx = 0.

U = Ax³ + Bx² − Cx + D

U = 3.65x³ + 3.25x² − 2.1x + 3.6

dU/dx = 3*(3.65)x² + 2*(3.25)x - 2.1 + 0

dU/dx = 10.95x² + 6.5x - 2.1

10.95x² + 6.5x - 2.1 = 0

Solving the above quadratic equation yields,

X₁ = -0.826

X₂ = 0.232

The stable point is the one for which the second derivative dU²/dx² > 0 and the unstable point is the one for which dU²/dx² < 0.

dU/dx = 10.95x² + 6.5x - 2.1

dU²/dx² = 2*(10.95)x + 6.5 - 0

dU²/dx² = 21.9x + 6.5

For X₁ = -0.826

dU²/dx² = 21.9(-0.826) + 6.5

dU²/dx² = -18.09 + 6.5

dU²/dx² = -11.59

Since dU²/dx² < 0, X₁ = -0.826 is the unstable point.

For X₂ = 0.232

dU²/dx² = 21.9(0.232) + 6.5

dU²/dx² = 5.08 + 6.5

dU²/dx² = 11.58

Since dU²/dx² > 0, X₂ = 0.232 is the stable point.

Answer 2

`x \approx 0.232\,m`.

Step-by-step explanation:

First and second derivatives of the potential energy function are, respectively:

`(dU)/(dx) = 3\cdot A\cdot x^(2) + 2\cdot B\cdot x - C`

`(d^(2)U)/(dx^(2)) = 6\cdot A \cdot x + 2\cdot B`

Particular forms of the function and its derivatives are:

`U = 3.65\cdot x^(3)+3.25\cdot x^(2)-2.1\cdot x+ 3.6`

`(dU)/(dx) = 10.95\cdot x^(2) + 6.5\cdot x - 2.1`

`(d^(2)U)/(dx^(2)) = 21.9 \cdot x + 6.5`

Let find the roots associated with the first derivative by using the general formula of the second-orden polynomial:

`10.95\cdot x^(2) + 6.5\cdot x - 2.1 = 0`

`x_(1)\approx 0.232\,m, x_(2)\approx -0.825`

The values of the second derivatives evaluated in each solution are:

`(d^(2)U)/(dx^(2)) = 21.9 \cdot (0.232\,m) + 6.5`

`(d^(2)U)/(dx^(2)) = 11.581`

`(d^(2)U)/(dx^(2)) = 21.9 \cdot (-0.825) + 6.5`

`(d^(2)U)/(dx^(2)) = -11.568`

`x_(1)` has an stable equilibrium solution. Hence,
`x \approx 0.232\,m`.