(a) take u = 6 for a system with a particle at position at x = 0 and calculate the potential energy as a function of the particle position x. the force on the particle is given by (8e-5x)î. u(x) = (b) explain whether the force is conservative or nonconservative and how you can tell.

the force on the particle is conservative, U(x) = 6 -8/5 e-5x

Explanation:

since the work done to go from any point (a,y1,z1) to (x,y2,z2) is:

W= ∫F dx = ∫8e-5x dx = 8 (-1/5)(e-5x - e-5a) = (-8/5) e-5x - (-8/5) e-5a

consequently, the work done for any trajectory that goes from (a,y1,z1) until (x,y2,z2) , where "a" is the starting point until "x" depends only on the values of a and x and not from the pathway taken, the force is conservative.

therefore

W(a= U(x,y2,z2) - U(a,y1,z1) , since the work done depends only of starting and final points (the integration limits).

taking a=0 and U=6 for the reference state

U(x) - 6 = 8 (-1/5)(e-5x - 0) → U(x) = 6 -8/5 e-5x

search it up!

explanation:

the closer the planet is to the sun, the larger the gravity