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+ <h3> Statement </h3>
+ <p>
+ $3.1.2.$ A weight of mass $m$ is suspended on a spring of stiffness $k$. How does the total force acting on the load depend on its displacement $x$ from the equilibrium position? Find the dependence of the potential energy of the load on its displacement $x$.
+ </p>
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+ <h3>Solution</h3>
+ <p>
+ The only external force other than the elastic force of the spring
+$$ \boxed{F=-kx} $$
+Let's find the change in energy when the spring moves by $dx$
+$$ dU=Fdx=kxdx $$
+After integrating $dU$ from $0$ to $U$
+$$ U=\int_{0}^{U}dU=k\int_{0}^{x}xdx=\fbox{$\frac{kx^2}{2}$} $$
+
+ </p>
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+ <h4>Answer</h4>
+ <p>
+ $$F=-kx;~U=\frac{kx^2}{2}$$
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