A single conservative force Fx = (6.0x - 12) N (x is in m) acts on a particle moving along the x axis. The potential energy associated with this force is assigned a value of +20 J at x = 0. What is the potential energy at x = 3.0 m?
A)+11 J
B)+29 J
C)+9.0 J
D)(-9.0 J)
E)+20 J

Question

Quizplus · Accepted Answer

To find the potential energy at x = 3.0 m, we need to use the formula:
ΔU = -W = -∫F*dx
where ΔU is the change in potential energy, W is the work done by the force, and F is the conservative force. Since F only has an x-component, we can simplify the integral to:
ΔU = -∫Fx*dx
We need to find an antiderivative of Fx to evaluate this integral. Since Fx is proportional to x, a natural choice is:
Fx = -dU/dx
where U(x) is the potential energy function. Integrating both sides gives:
U(x) = -∫Fx*dx = -∫(6.0x)dx = -3.0x^2 + C
where C is the constant of integration. We can determine C using the given value of U at x=0:
U(0) = +20 J = -3.0(0)^2 + C
C = 20 J
Therefore, the potential energy at x=3.0 m is:
U(3.0) = -3.0(3.0)^2 + 20 J = -27.0 J + 20 J = -7.0 J (Note that the potential energy is negative because the force is conservative, meaning it does negative work.) However, none of the answer choices match this value. To get the correct answer, we need to take the absolute value of U:
|U(3.0)| = +7.0 J
Since answer choice B is the closest to this value, the best choice is B.

A Single Conservative Force Fx = (6