The potential energy for a certain mass moving in one dimension is given by U(x) = (2.0 J/m^3)x^3 - (15 J/m^2)x^2 + (36 J/m)x - (23 J). Find the location(s) where the force on the mass is zero.
The potential energy is given by
`U(x) = 2x^3 - 15x^2 + 36x - 23`
The force with which this potential energy is associated is
`F = -(dU)/(dx)`
Find the derivative of U with respect to x:
`(dU)/(dx) = 6x^2-30x + 36`
The force is zero when the function above is 0, that is,
`6x^2 - 30x + 36 = 0`
Divide by 6:
`x^2 - 5x + 6 = 0`
Solve by factoring:
(x-2)(x-3) = 0
x = 2 and x = 3
At x = 2 and x = 3 the force on the mass will be 0.