Nonisolated system?? (block and spring)?

Question

A block of mass 12 kg slides from rest down a frictionless 35 deg incline and is stopped by a strong spring with a force constant of (3 * 10^4 N/m). The block slides 3m from rest against the spring. When the block comes to rest, how far has the spring been compressed?

Eep! How do I figure this out?

Answer 1

I assume that the total travel along the incline including the spring compression is L = 3m. The potential energy lost in this slide is m*g*h, where h = vertical distance covered. This is L*sinø, where ø is the 35º angle. So the total energy gained by the block and then absorbed by the spring is m*g*L*sinø. The energy absorbed by the spring is given by E = ∫F*ds, where F is the force from the spring. If the spring constant is k, then F=ks, and E = k∫s*ds [0 to ∆s] = .5*k*∆s^2. ∆s is the compression of the spring. Equate this to the energy gained by the block to get

m*g*L*sinø = .5*k*∆s^2

∆s^2 = 2*[m*g*L*sinø]/k

∆s = √[2*(m*g*L*sinø)/k]

Answer 2

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