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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> |
| $3.1.7.$ A horizontal trough to the left of the bottom line is bent on a cylindrical surface of radius $r$, and to the right, on a surface of radius $R$. Determine the ratio of the largest deflections to the left and to the right during small oscillations of the body in this trough. | | $3.1.7.$ A horizontal trough to the left of the bottom line is bent on a cylindrical surface of radius $r$, and to the right, on a surface of radius $R$. Determine the ratio of the largest deflections to the left and to the right during small oscillations of the body in this trough. |
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| loading="lazy" width="150" /> | | loading="lazy" width="150" /> |
| <figcaption> | | <figcaption> |
| For problem $3.1.7$ | | For problem $3.1.7$ |
| </figcaption> | | </figcaption> |
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| </center> | | </center> |
| <p> | | <p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| According to the law of conservation of body energy, the maximum potential energy will be conserved in both cases | | According to the law of conservation of body energy, the maximum potential energy will be conserved in both cases |
| $$ E_{p1}=E_{p2} $$ | | $$ E_{p1}=E_{p2} $$ |
| And therefore the height $h$ will rise relative to the lower point of the trajectory to which the body will rise | | And therefore the height $h$ will rise relative to the lower point of the trajectory to which the body will rise |
| will be equal | | will be equal |
| | | |
| </p> | | </p> |
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| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/3/3.1.7/3.1.7_1.png" | | <img src="https://savchenkosolutions.com/3/3.1.7/3.1.7_1.png" |
| loading="lazy" width="230" /> | | loading="lazy" width="230" /> |
| <figcaption> | | <figcaption> |
| The position of the ball at the extreme points of the trajectory | | The position of the ball at the extreme points of the trajectory |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| We will use the expression $\Delta U$ for small oscillations obtained in <a href="../3.1.5">3.1.5</a> | | We will use the expression $\Delta U$ for small oscillations obtained in <a href="../3.1.5">3.1.5</a> |
| $$ \Delta U = \frac{mgx_1^2}{2r}=\frac{mgx_2^2}{2R} $$ | | $$ \Delta U = \frac{mgx_1^2}{2r}=\frac{mgx_2^2}{2R} $$ |
| Whence it follows that | | Whence it follows that |
| $$ \boxed{\frac{x_2}{x_1}=\sqrt{\frac{R}{r}}} $$ | | $$ \boxed{\frac{x_2}{x_1}=\sqrt{\frac{R}{r}}} $$ |
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| </p> | | </p> |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$\frac{x_2}{x_1}=\sqrt{\frac{R}{r}}$$ | | $$\frac{x_2}{x_1}=\sqrt{\frac{R}{r}}$$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Dzikan Mikita<br> | | Dzikan Mikita<br> |
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