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I am a high school student wondering if the pulley system described below will work, and if so, how to go about it.

Basically, I have a small system where as you pull back an object, it causes two lengths, labeled $d$ and $p$, to move forward. $d$ is dependent on many factors other than just the pulley system, but $p$ is just growing linearly as the object is pulled back. I've determined that the relationship between $d$ and $p$ is: $$p=\sqrt{A+Bd^2}-C$$ Where $A$, $B$, and $C$ are positive constants.

Here is a diagram to visualize how $p$ is connected to the system:

enter image description here

The length $p$ is the distance from the smaller block to the edge of the base. As you pull the object back, the string is pulled which pulls the spring and increases the length $p$. My question is, is there any sort of pulley system or just general construction I can insert between the object you pull and the pulley which is connected to the smaller block such that as you pull the object back, $p$ does not increase at a linear rate, but instead increases at the exact same rate of $d$.

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    $\begingroup$ I don't see anything labeled "d" in the image. Without that, I'm not sure what your problem is. $\endgroup$ – hazzey♦ May 8 '15 at 13:15
  • $\begingroup$ Where is d in this diagram? All you have is a rope going around a pulley, which ends at a spring attached to the wall. Everything is linearly dependent on how far you pull the rope. The spring stretches by exactly how far you pull the rope, which is what increases p in your diagram. $\endgroup$ – Nuclear Hoagie May 8 '15 at 13:17
  • $\begingroup$ $d$ is not in the diagram; $d$ is a different variable that arises in a different manner. Basically after the object is pulled back, a pendulum is released which hits a ball and sends it flying, and the distance away from its original position is $d$. $\endgroup$ – ASKASK May 8 '15 at 14:07
  • $\begingroup$ What is the magnitude of A relative to Bd^2? $\endgroup$ – Air May 8 '15 at 18:35
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    $\begingroup$ You must have some idea of their magnitudes, if this is a real physical system. In any case, some more practical details would be really helpful. It matters what range of motion you need, what the constants represent, what you're actually trying to do... You say "d" is dependent on other factors, but then you give an equation that indicates "d" depends directly on "p" and no other variables. $\endgroup$ – Air May 8 '15 at 23:00
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If I understand your question correctly, then you want to know if there is some mechanism which can convert a movement of distance $d$ to a movement of distance $p$, given that you want $p=\sqrt{A+Bd^2}-C$. If $d$ is caused by pulling on a rope, then the easiest way I can think to achieve your objective is a spring-loaded cam. You would need to relate $d$ to the angle of turn of the cam and then make the radius of the cam the appropriate function of angle.

However, this depends on you having a low requirement for range of motion. You could of course, use a block and tackle to amplify the movement, but you will run up against size and frictional limitations. Additionally, it may be difficult to fabricate your cam.

I don't know the details of your project, but it might be worthwhile to look at the range of $d$ and $p$ that you expect and see if you can come up with a linear relation that approximates your desired relation to the desired degree of accuracy. It is much easier to design linear mechanisms and you may find that your equation is needlessly complex.

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