A climbing rope exerts a force given by F=−kx−cx2. Find an expression for c such that when the rope is stretched a distance d its potential energy is twice what it would be if the rope were an ideal spring with F=−kx.

Explanation:

For the climbing rope, the force is given by,

Now, we know that for any conservative force,

(one dimensional case)

where, V = potential energy

Therefore, the potential energy of the rope is given by,

a is a constant, which can be taken to be zero depending on our choice of zero potential. Hence,

Similarly, we can find the potential energy of an ideal spring,

Now, it is given that at x = d,

or,

or,

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the electric field is a “force field” around a charged object that illustrates the direction the electric force would push an imaginary