Question

A force F(x) = (−5.0x2 + 4.0x) N acts on a particle. (a) How much work (in J) does the force do on the particle as it moves from x = 4.0 m to x = 6.0 m? J (b) Picking a convenient reference point of the potential energy to be zero at x = 0, find the potential energy for this force. (Use the following as necessary: x. Assume U(x) is in joules and x is in meters. Do not include units in your answer.)

Answer 1

The work done by the force as it moves from x = 4.0 m to x = 6.0 m is calculated by integrating the force function over the interval. The potential energy of the system is found by integrating the negative of the force and applying the boundary condition.

Step-by-step explanation:

To answer the student's question:

(a) The work done by the force F(x) = (-5.0x2 + 4.0x) N as it moves from x = 4.0 m to x = 6.0 m is found by integrating F(x) over the given interval. The integral of F(x) from 4.0 m to 6.0 m gives us the work:

W = ∫4.06.0 F(x) dx

(b) To find the potential energy (U), we integrate the negative of the force function, since U = -W. The potential energy relative to a convenient reference point of U = 0 at x = 0 is:

U(x) = - ∫ F(x) dx + C

After integration, we apply the boundary condition U(0) = 0 to solve for the constant C.