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| Solutions of Savchenko Problems in Physics <br> | | Solutions of Savchenko Problems in Physics <br> |
| <i><b>knowledge must be free</b></i> | | <i><b>knowledge must be free</b></i> |
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| <h3 id="back-link"><a href="../../#2.7">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.7">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.7.44.$ The moment of inertia of a solid body of mass $m$ relative to the $O$-axis is equal to $J$. The center of mass of the body is located at a distance $R$ from this axis. Find the force acting on the axis when a force $F$ is briefly applied to a solid body perpendicular to the segment of length $x$ that connects the point of application of the force and the axis. At which $x$ is the smallest force acting on the axis? | | $2.7.44.$ The moment of inertia of a solid body of mass $m$ relative to the $O$-axis is equal to $J$. The center of mass of the body is located at a distance $R$ from this axis. Find the force acting on the axis when a force $F$ is briefly applied to a solid body perpendicular to the segment of length $x$ that connects the point of application of the force and the axis. At which $x$ is the smallest force acting on the axis? |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.7.44.png" | | <img src="2.7.44.png" |
| loading="lazy" width="130" /> | | loading="lazy" width="130" /> |
| <figcaption> | | <figcaption> |
| For problem $2.7.44$ | | For problem $2.7.44$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| | | |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.7.44_1.png" | | <img src="2.7.44_1.png" |
| loading="lazy" width="130" /> | | loading="lazy" width="130" /> |
| <figcaption> | | <figcaption> |
| Force $\vec{F}_0$ that will produce the rotational torque | | Force $\vec{F}_0$ that will produce the rotational torque |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| $$F-F_0=m\varepsilon R$$ | | $$F-F_0=m\varepsilon R$$ |
| $$F_0=F-mR\cdot \frac{Fx}{J}\Leftrightarrow \boxed{F_0=F\left(1-\frac{mRx}{J}\right)}$$ | | $$F_0=F-mR\cdot \frac{Fx}{J}\Leftrightarrow \boxed{F_0=F\left(1-\frac{mRx}{J}\right)}$$ |
| From here, the minimum values of $F_0 = 0$, described by the expression | | From here, the minimum values of $F_0 = 0$, described by the expression |
| $$F_0=0\Leftrightarrow 1-\frac{mRx}{J}=0$$ | | $$F_0=0\Leftrightarrow 1-\frac{mRx}{J}=0$$ |
| which will be observed at length that connects the point of application of the force and the axis | | which will be observed at length that connects the point of application of the force and the axis |
| $$\boxed{x=\frac{J}{mR}}$$ | | $$\boxed{x=\frac{J}{mR}}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $F'=F(1-mRx/J)$. With $x=J/(mR),~F'=0$ | | $F'=F(1-mRx/J)$. With $x=J/(mR),~F'=0$ |
| </p> | | </p> |
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| Almaskhan Arsen<br> | | Almaskhan Arsen<br> |
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