Find the force corresponding to the potential energy U(x) = (Use the following as necessary: a, b, and x. Express your answer in vector form.)

Consider a force in R3 defined by f(x,y,z)=(6*x*y^3, 9*x^2*y^2+2*y,0). Find the potential energy U(x,y,z) such that U(0,0,0)=0.

Please explain  An electrically charged particle is held fixed at the origin. Let V = 1/r be the potential energy of a second particle with unit electric charge located at (x, y, z), where r = sqrt(x^2 + y^2 + z^2). The force on this second particle is given by −∇V . Calculate the force on the particle when it is at (1, 2, −2). Show working

A box with weight of magnitude FG=2.00N is lowered by a rope down a smooth plane that is inclined at an angle ϕ = 30.0 ∘ above the horizontal, as shown in (Figure 1) at left. The normal force acting on the box has a magnitude n=1.73N the tension force is 1.00 N, and the displacement Δr of the box is 1.80 m down the inclined plane. The work done on the box by gravity is 1.8 J What is the work Wn done on the box by the normal force? What is the work WT done by the tension force?

a.) A particle with mass m is dropped with zero initial velocity a height h above the ground, under the action of the gravitational field (acceration constant g), such that it reaches the ground (z=0) with zero potential energy and velocity v>0. Determine h, in terms of the other quantities given. b.) Another particle moves with Simple Harmonic Motion with centre O.The particle has velocity 13ms–1 when it is 3m from O and 5ms–1 when it is 5m from O.(i) Find the period and amplitude of the motion

is the vector foeld conservative? prove it.

I need the answer as soon as possible