The potential energy of an object as a function of position is U(x)= x³ + 8,where U is measured in joules and x is measured in meters.Where is the object be located if it is in a metastable equilibrium?

Question

Quizplus · Accepted Answer

For an object to be in metastable equilibrium, the first derivative of its potential energy with respect to position (dU/dx) should be zero, and the second derivative (d²U/dx²) should be greater than zero. The given potential energy function is U(x) = x^13 + 8. The first derivative with respect to x is dU/dx = 13x^12. Setting this equal to zero, we find that x = 0 is the only point where the derivative is zero, indicating a potential equilibrium point. The second derivative is d²U/dx² = 156x^11, which is positive for all x except x = 0, where it is exactly zero. However, since there's no other point where the first derivative is zero, x = 0 is the closest condition to a metastable equilibrium given the options, even though strictly speaking, a positive second derivative would be preferred to confirm stability.

The Potential Energy of an Object as a Function of Position