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?

Savchenko Solutions

Solutions of Savchenko Problems in Physics
knowledge must be free

    <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?

    <h3>Solution</h3>
    <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 $$\tau_0=\tau_f=Fx=J\varepsilon$$ From where $$\varepsilon = \frac{Fx}{J}$$ Newton's second law for the rotational motion $$F+F_0=m\varepsilon R$$ $$F_0=mR\cdot \frac{Fx}{J}-F\Leftrightarrow \boxed{F_0=F\left(\frac{mRx}{J}-1\right)}$$ From here, the minimum values of $F_0 = 0$, described by the expression $$F_0=0\Leftrightarrow \frac{mRx}{J}-1=0$$ which will be observed at length that connects the point of application of the force and the axis $$\boxed{x=\frac{J}{mR}}$$

    <h4>Answer</h4>
    <p>
        $F'=F(mRx/J-1)$. With $x=J/(mR),\~F'=0$
    </p>


<footer class="row container">
  <br>
    <p>
        <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small>
    </p>
    <p>
        <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>
    </p>
</footer>