Making inseparable words in the heading @ GUI
en/2.7.44.md
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| <meta name="author" content="Aliaksandr Melnichenka"> | |||
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| <meta property="og:title" content="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?"> | |||
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| <title>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?</title> | |||
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| @@ -39,7 +39,7 @@ | |||
| <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | |||
| </div> | |||
| <p class="author"> | |||
| − | Solutions | ||
| + | Solutions of Savchenko Problems in Physics <br> | ||
| <i><b>knowledge must be free</b></i> | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#2.7">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <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? | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="2.7.44.png" | |||
| loading="lazy" width="130" /> | |||
| <figcaption> | |||
| For problem $2.7.44$ | |||
| @@ -59,8 +59,6 @@ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| − | <p> | ||
| − | </p> | ||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="2.7.44_1.png" | |||
| loading="lazy" width="130" /> | |||
| <figcaption> | |||
| Force $\vec{F}_0$ that will produce the rotational torque | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| Force $\vec{F}$ creates an angular momentum $\vec{F}\cdot x$ (with $x$ perpendicular to $\vec{F}$): | |||
| $$\tau_0=F\cdot x$$ | |||
| $$\tau_f=J\cdot\varepsilon$$ | |||
| In steady-state mode, the angular moments will be equal | |||
| @@ -83,18 +81,21 @@ | |||
| $$\tau_0=\tau_f=Fx=J\varepsilon$$ | |||
| From where | |||
| $$\varepsilon = \frac{Fx}{J}$$ | |||
| − | Newton's second law | ||
| − | $$F | ||
| − | $$F_0=mR\cdot \frac{Fx}{J} | ||
| + | Newton's second law | ||
| + | $$F-F_0=m\varepsilon R$$ | ||
| + | $$F_0=F-mR\cdot \frac{Fx}{J}\Leftrightarrow \boxed{F_0=F\left(1-\frac{mRx}{J}\right)}$$ | ||
| From here, the minimum values of $F_0 = 0$, described by the expression | |||
| − | $$F_0=0\Leftrightarrow \frac{mRx}{J} | ||
| + | $$F_0=0\Leftrightarrow 1-\frac{mRx}{J}=0$$ | ||
| which will be observed at length that connects the point of application of the force and the axis | |||
| $$\boxed{x=\frac{J}{mR}}$$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| − | $F'=F(mRx/J | ||
| + | $F'=F(1-mRx/J)$. With $x=J/(mR),~F'=0$ | ||
| + | </p> | ||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
| + | Almaskhan Arsen<br> | ||
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| <meta name="description" content="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?"> | <meta name="description" content="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?"> | ||
| <meta name="author" content="Aliaksandr Melnichenka"> | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| <meta property="og:title" content="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?"> | <meta property="og:title" content="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?"> | ||
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| <title>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?</title> | <title>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?</title> | ||
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| @@ -39,7 +39,7 @@ | |||
| <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | ||
| </div> | </div> | ||
| <p class="author"> | <p class="author"> | ||
| Solutions |
Solutions of Savchenko Problems in Physics <br> | ||
| <i><b>knowledge must be free</b></i> | <i><b>knowledge must be free</b></i> | ||
| </p> | </p> | ||
| </header> | </header> | ||
| <h3 id="back-link"><a href="../../#2.7">$\leftarrow$Back</a></h3> | <h3 id="back-link"><a href="../../#2.7">$\leftarrow$Back</a></h3> | ||
| <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$ | ||
| @@ -59,8 +59,6 @@ | |||
| </figcaption> | </figcaption> | ||
| </figure> | </figure> | ||
| </center> | </center> | ||
| <p> | |||
| </p> | |||
| <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> | ||
| <p> | <p> | ||
| Force $\vec{F}$ creates an angular momentum $\vec{F}\cdot x$ (with $x$ perpendicular to $\vec{F}$): | Force $\vec{F}$ creates an angular momentum $\vec{F}\cdot x$ (with $x$ perpendicular to $\vec{F}$): | ||
| $$\tau_0=F\cdot x$$ | $$\tau_0=F\cdot x$$ | ||
| $$\tau_f=J\cdot\varepsilon$$ | $$\tau_f=J\cdot\varepsilon$$ | ||
| In steady-state mode, the angular moments will be equal | In steady-state mode, the angular moments will be equal | ||
| @@ -83,18 +81,21 @@ | |||
| $$\tau_0=\tau_f=Fx=J\varepsilon$$ | $$\tau_0=\tau_f=Fx=J\varepsilon$$ | ||
| From where | From where | ||
| $$\varepsilon = \frac{Fx}{J}$$ | $$\varepsilon = \frac{Fx}{J}$$ | ||
| Newton's second law |
Newton's second law | ||
| $$F |
$$F-F_0=m\varepsilon R$$ | ||
| $$F_0=mR\cdot \frac{Fx}{J} |
$$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 \frac{mRx}{J} |
$$F_0=0\Leftrightarrow 1-\frac{mRx}{J}=0$$ | ||
| 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(mRx/J |
$F'=F(1-mRx/J)$. With $x=J/(mR),~F'=0$ | ||
| </p> | |||
| <p style="text-align: right; font-style: italic; font-size: 14;"> | |||
| Almaskhan Arsen<br> | |||
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| <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small> | <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small> | ||
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