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,

sleep paralysis is a parasomnia, events associated with sleep.

sleep paralysis happens when you are in the middle of waking up or falling asleep. it is the feeling of being conscious but unable to move. it definitely involves hallucinations.

i had a friend that had sleep paralysis and he told me a story one time where he was laying in bed and he couldn't move his body no matter how hard he tried and there was someone standing in the corner of his room coming towards him. he couldn't move and you can't "wake up" because you are already conscious. sometimes sleep paralysis can be triggered by frequent waking up during the night based off of multiple trips to the bathroom, or if you had ever been abused, sleep paralysis can start happening from things like that.

the answer would be 32*f