Conservation of Energy

Question

The potential energy function of a conservative force acting on an object is given by the expression

U=y-2x²-2x

where U is in joules and x, y, z are in meters. Consider the following two points in space:

A=(1 m,1 m,1 m);

B=(2 m,2 m,2 m);

(a) Find the value of the force F that acts when the object is at the point A.

(b) Suppose that the object moves from A to B along some unspecified path. What is the work done by the force during that motion?

(c) Suppose that the only force that acts on the object is this conservative force. Find the changes in the kinetic energy, ΔK, and the total mechanical energy, ΔE of the object for the motion in part (b).

Answer 1

(a)

The force F is the negative gradient of the potential energy U:

F = (4x+2, -1, 0)

At point A:

F = (6, -1, 0)

(b)

In general, work W is the integral of the force F over a path.

I do not know whether you are expected to actually compute the integral;

if so you can pick any convenient path, as the force is conservative.

Otherwise, the work is simply the difference in potential energy:

W = U(A) - U(B)

Note the sign: any decrease of potential energy when going from A to B

implies positive work.

Substituting actual values:

W = (1 - 2 - 2) - (2 - 8 - 4) = 7J

(c)

By conservation of energy, ΔE = 0,

and ΔK = 7J, namely the reduction in potential energy calculated in (b).