A conservative force on a particle moving along the x axis is given by $\overrightarrow { \mathbf { F } } = \left( 3 x ^ { 2 } - 2 x \right) \tilde{ \mathbf { i } }$ .Which of the following is a potential that is associated with this force?
A) $( 6 x - 2 ) \tilde{ \mathbf { i } }$
B) $( - 6 x + 2 ) \tilde{ \mathbf { i } }$
C) $x ^ { 3 } - x ^ { 2 } + 3$
D) $- x ^ { 3 } + x ^ { 2 } + 3$
E)No answer given above is correct.

Question

Quizplus · Accepted Answer

The potential energy function $ U(x) $ is related to the force by $ F(x) = -\frac{dU}{dx} $. Integrating the given force $ F(x) = 3x^2 - 2x $ with respect to $ x $ gives $ U(x) = -\left(\frac{3}{3}x^3 - \frac{2}{2}x^2\right) + C = -x^3 + x^2 + C $, which matches option D.

A Conservative Force on a Particle Moving Along the X F→=(3x2−2x)i~\overrightarrow { \mathbf { F } } = \left( 3 x ^ { 2 } - 2 x \right) \tilde{ \mathbf { i } }F=(3x2−2x)i~

A Conservative Force on a Particle Moving Along the X $\overrightarrow { \mathbf { F } } = \left( 3 x ^ { 2 } - 2 x \right) \tilde{ \mathbf { i } }$