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| <h3 id="back-link"><a href="../../#3.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#3.1">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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 $x$. | | $3.1.2.$ 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> | | </p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| The only external force other than the elastic force of the spring | | The only external force other than the elastic force of the spring |
| $$ \boxed{F=-kx} $$ | | $$ \boxed{F=-kx} $$ |
| Let's find the change in energy when the spring moves by $dx$ | | Let's find the change in energy when the spring moves by $dx$ |
| $$ dU=Fdx=kxdx $$ | | $$ dU=Fdx=kxdx $$ |
| After integrating $dU$ from $0$ to $U$ | | After integrating $dU$ from $0$ to $U$ |
| $$ U=\int_{0}^{U}dU=k\int_{0}^{x}xdx=\fbox{$\frac{kx^2}{2}$} $$ | | $$ U=\int_{0}^{U}dU=k\int_{0}^{x}xdx=\fbox{$\frac{kx^2}{2}$} $$ |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
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| $$F=-kx;~U=\frac{kx^2}{2}$$ | | $$F=-kx;~U=\frac{kx^2}{2}$$ |
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