Determine the force of the tension of the thread connecting two balls of volume 10 cm^3 each, if the upper ball floats half-immersed in water. The lower ball is three times heavier than the upper one.
<h3 id="back-link"><a href="/#4.2">$\leftarrow$Back</a></h3>
<h3> Statement </h3>
<p>
$4.2.12$
Determine the force of the tension of the thread connecting two balls of volume 10 cm$^3$ each, if the upper ball floats half-immersed in water. The lower ball is three times heavier than the upper one.
</p>
<center>
<figure>
<img src="statement.png"
loading="lazy" alt="4.2.12" width="200" />
<figcaption>
For problem 4.2.12
</figcaption>
</figure>
</center>
<h3>Solution</h3>
<p>
Applying Newton's Second Law...
</p>
<center>
<figure>
<img src="draw.png"
loading="lazy" alt="4.2.12" width="200" />
<figcaption>
Force analysis
</figcaption>
</figure>
</center>
<p>
For upper ball
$$m_{\uparrow}\~g+T = F_{\uparrow}$$
$$m_{\uparrow}\~g+T = \rho_0\~g\~\frac{V}{2} \\;(1)$$
For lower ball
$$m_{\downarrow}\~g = T+F_{\downarrow}$$
$$m_{\downarrow}\~g = T+\rho_0\~g\~V \\;(2)$$
Separating the gravity's force term from $(1)$
$$m_{\uparrow}\~g = \rho_0\~g\~\frac{V}{2}-T \\;(3)$$
Dividing $(2)$ by $(3)$, and taking account that $m_{\downarrow}=3\~m_{\uparrow}$
$$\frac{\rho_0\~g\~V+T}{\rho_0\~g\~\frac{V}{2}-T}=3$$
and separating $T$
$$T = \frac{\rho_0\~g\~V}{8}$$
Taking $\rho_0 = 1\~\rm{g/cm^3} = 10^{-3}\~\rm{kg/cm^3}$ (water's density) and $g=9.8\~\rm{m/s^2}$ (acceleration due to gravity at Earth's surface),
$$\boxed{T \simeq 1.2 \cdot 10^{-2}\~\rm{N}}$$
</p>
<p style="text-align: right; font-style: italic; font-size: 14;">
BSc. Luis Daniel Fernández Quintana<br>
Physics Department (FCNE)<br>
Universidad de Oriente, Cuba<br>
</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>
Savchenko Solutions