Measurements of the circular frequency of oscillations of a body of mass m, fixed in the middle of a tensioned string with length 2 l, gave the value \omega. Find the force of tension of the string.

Savchenko Solutions

Solutions of Savchenko Problems in Physics
knowledge must be free

    <h3 id="back-link"><a href="/#3.2">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $3.2.10.$ Measurements of the circular frequency of oscillations of a body of mass $m$, fixed in the middle of a tensioned string with length $2 l$, gave the value $\omega$. Find the force of tension of the string.
    </p>

    <h3>Solution</h3>
    <p>
        The total external force acting on the system

$$F=2T\sin\varphi$$
Newton's second law
$$m\ddot{x}(t)+2T\sin\varphi=0$$
Let's use the approximation for small angles $(\varphi \ll 1)$:
$$\sin\varphi\approx\varphi=\frac{x}{l}$$
Harmonic oscillation equation
$$\ddot{x}(t)+\frac{2T}{ml}x(t)=0$$
Where does the period of oscillation come from?
$$\omega^{*2}=\frac{2T}{ml}\Rightarrow \boxed{T=\frac{ml\omega^{*2}}{2}}$$

Dzikan Mikita

Aliaksandr Kanashenka

    <h4>Answer</h4>
    <p>
        $$T=\frac{ml\omega^{*2}}{2}$$
    </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>