The potential energy for a certain mass moving in one dimension is given by U(x)= (2.0 J/m³)x³- (15 J/m²)x²+ (36 J/m)x - 23 J.Find the location(s)where the force on the mass is zero. A)4.0 m,5.0 m B)1.0 m C)2.0 m,3.0 m D)3.0 m,5.0 m

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Quizplus · Accepted Answer

The force on the mass is given by the negative gradient of the potential energy, $ F = -\frac{dU}{dx} $. For the given potential energy function $ U(x) = 2x^3 - 15x^2 + 36x - 23 $, the force is zero where the derivative of $ U(x) $ with respect to $ x $ is zero. Differentiating $ U(x) $ gives $ \frac{dU}{dx} = 6x^2 - 30x + 36 $. Setting this equal to zero and solving for $ x $ gives $ x = 2 $ and $ x = 3 $, which are the points where the force on the mass is zero.

The Potential Energy for a Certain Mass Moving in One