
Ok, here we go..
Assumption: weight distribution is 50:50 (ie 90kg/wheel)
Static Equilibrium => summing moments about any point in space will equal zero. I will choose the swing arm hinge as my point of reference.
Moment due to weight = opposing moment due to spring
cos(50)(.570m)(90kg) = sin(25)(0.1425m)(spring weight force)
Spring weight force = (cos(50)(.570m)(90kg)) / (sin(25)(0.1425m)) = 547.5kg
So the spring must withstand 547.5kg (or 5371N)
Hookes law:
F = kx
We have F and we have two unknowns; k and x, we have used all the given information, we are stuck.. without knowing either k or x we cannot calculate the other and we have no information at all about k or x...
We can plug some random figures in:
if we say k is 15kg/mm:
x = F/k = (547.5kg)/ (15kg/mm) = 36.5mm (of spring compression)
If we say x is 20mm
k = F/x k = (547.5kg)/(20mm) = 27.4kg/mm (spring constant)
Your spring rate is the least of your problems though, your swing arm is closer to being vertical than it is to horizontal (50 degrees from horizontal!!)

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