385. When a spring is stretched by 5 cm, the work expended is 29.43 J. By how much will the spring stretch if the work expended is 9.81 J?
Solution. Here $A_{1}=29.4$ J, $x_{0}=0, x_{1}=0.05$ m. Since $A_{1}=$ $=k \int_{x_{0}}^{x_{1}} x d x$, we obtain the equation $29.43=k \int_{0}^{0.05} x d x \quad$ or $\quad 29.43=k \times$ $\times\left.\frac{x^{2}}{2}\right|_{0} ^{0.05}$, from which $k=\frac{29.43 \cdot 2}{0.0025}=23544$.

Next, from the equation $A_{2}=k \int_{x_{0}}^{x_{2}} x d x$, where $A_{2}=9.81$ J, $x_{0}=0, k=$ $=23544$, we find $x_{2}$. We have $9.81=\left.23544 \frac{x^{2}}{2}\right|_{0} ^{x_{2}}$, from which $x_{2}^{2}=\frac{9.81 \cdot 2}{23544}=$ $=0.00083$, i.e., $x_{2}=\sqrt{0.00083} \approx 0.029$ (m).