Answer:
[tex]k = \frac{2\cdot m \cdot g \cdot (d+x_{f})\cdot (\sin \theta - \mu_{k}\cdot \cos \theta)}{x_{f}^{2}}[/tex]
Explanation:
Let assume that spring reaches its maximum compression at a height of zero. The system is modelled after the Principle of Energy Conservation and the Work-Energy Theorem:
[tex]U_{g,A}=U_{k,B} + W_{f}[/tex]
[tex]m\cdot g \cdot (d + x_{f})\cdot \sin \theta = \frac{1}{2}\cdot k \cdot x_{f}^{2}+\mu_{k}\cdot m \cdot g \cdot (d+x_{f})\cdot \cos \theta[/tex]
[tex]m\cdot g \cdot (d + x_{f})\cdot (\sin \theta-\mu_{k}\cdot \cos \theta) = \frac{1}{2}\cdot k \cdot x_{f}^{2}[/tex]
The spring constant is cleared in the expression described above:
[tex]k = \frac{2\cdot m \cdot g \cdot (d+x_{f})\cdot (\sin \theta - \mu_{k}\cdot \cos \theta)}{x_{f}^{2}}[/tex]
