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,

when earth is in rotational motion around the sun,it displaces from its place.

when a ball is placed on an inclined surface it does both the rotational and circular motion.

explanation:

the force on the thing and the distance the force moves the thing