Answer:
[tex]\displaystyle U=-3.45x^2-3.3x^3[/tex]
[tex]F(0.265)=-2.52\ N[/tex]
Explanation:
Spring Potential Energy
For any expression of the restoring force F as a function of the displacement from the equilibrium point x, the change of potential energy is equivalent to the work done to move the object from one point to another.
(a) The potential energy is
[tex]\displaystyle U=\int_{0}^{x}F(x).dx[/tex]
We have that
[tex]F(x)=-6.9x-9.9x^2[/tex]
Thus
[tex]\displaystyle U=\int_{0}^{x}(-6.9x-9.9x^2).dx[/tex]
[tex]\displaystyle U=-6.9\frac{x^2}{2}-9.9\frac{x^3}{3}[/tex]
[tex]\displaystyle U=-3.45x^2-3.3x^3[/tex]
(b) Knowing that x=26.5 cm = 0.265 m
[tex]F(0.265)=-6.9\cdot 0.265-9.9\cdot 0.265^2[/tex]
[tex]F(0.265)=-2.52\ N[/tex]
