In order to receive full credit on correct solutions and partial credit on incor

Question

In order to receive full credit on correct solutions and partial credit on incorrect solutions you must completely show your work (including the formulas used and the methodology employed) in the space provided. Consider a particle acted upon by a single, one-dimensional, conservative force described by the potential energy function U(x) = x^4 - 12x^2 where U(x) is in Joules and x is in meters. Part A. Determine the equilibrium positions of the particle Part B. Identify whether they are stable or unstable.

Explanation / Answer

A) At equilibrium positions, dU/dx = 0

=> 4x3 - 24x = 0

=> x(x2 - 6) = 0

=> x = -61/2 m, 0 m, 61/2 m = -2.45 m, 0 m, 2.45 m

B) If the second derivative (d2U/dx2) is negative, its an unstable equilibrium. If its positive, it is stable equilibrium.

d2U/dx2 = 12x2 - 24

At x = 0 m, d2U/dx2 = (12 * 02) - 24 = -24

So, x = 0 m is unstable equilibrium.

At x = -61/2 m, d2U/dx2 = [12 * (-61/2)2] - 24 = 48

So, x = -61/2 m is stable equilibrium.

At x = 61/2 m, d2U/dx2 = [12 * (61/2)2] - 24 = 48

So, x = 61/2 m is stable equilibrium.

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