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.

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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>

    <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>

    <h4>Answer</h4>
    <p>
        $$F=-kx;\~U=\frac{kx^2}{2}$$
    </p>
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