I'm in the process of trying to combine some equations for a simulation.
One describes the position at a point along a hanging slinky using...
$$y(d) = (l_1 + {mg\over k})d - {mg\over 2k}d^2$$ (source: http://large.stanford.edu/courses/2007/ph210/kolkowitz1/)
where...
- $d$ is a point between [0-1] along the spring's mass. ie .5 is the midway point, and the mass up until that point is total mass/2
- $l_1$ is the length of the slinky without any external force. Since a slinky is a 'pretensioned' spring, at rest its coils are touching. This is the 'minimum' length, and the slinky is displaced from here
- $mg\over k$ denotes the total spring displacement when hanging due to the force of gravity. This is derived from Hooke's law and Newton's 2nd law
- $k$ is the spring constant
Part 1 I understand:
$$(l_1 + {mg\over k})d$$
This just means the (minimum length + displacement) * fraction of mass. Since the mass is directly proportional to displacement, this should give us the displacement at point $d$ along the slinky spring
Part 2 I don't understand:
$$-{mg\over 2k}d^2$$
This is obviously (I think) the integral of ${mg\over k}d$. But why? ${mg\over k}d$ is not a rate, such as velocity, but it is a distance. And why do we subtract it from Part 1?