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<title>Two equal trunks are placed as the figure shows. The lower trunk is linked to the vertical wall by wires that form an angle of 45^\circ with it. The upper trunk is half submerged in water. Find the density of the trunks.</title>
Two equal trunks are placed as the figure shows. The lower trunk is linked to the vertical wall by wires that form an angle of 45$^\circ$ with it. The upper trunk is half submerged in water. Find the density of the trunks.
</p>
<center>
<figure>
<img src="statement.png"
loading="lazy" alt="4.2.13" width="300" />
<figcaption>
For problem 4.2.13
</figcaption>
</figure>
</center>
<h3>Solution</h3>
<p>
Let's analyze the following force diagram
</p>
<center>
<figure>
<img src="diagram.png"
loading="lazy" alt="4.2.13" width="300" />
<figcaption>
Force diagram for each trunk
</figcaption>
</figure>
</center>
<p>
Consider the trunks are perfect wooden cylinders. Applying Newton Second Law for cylinder 1, on $x$-direction,
$$-N'\cos{45^\circ}+N = 0 \;(1)$$
and on $y$-axis
$$mg = \rho_0 g V_s + N'\sin{45^\circ} = \rho_0 g V_{s1} + N'\frac{\sqrt{2}}{2}$$
as trunk 1 is half submerged, $V_{s1} = \frac{\pi R^2 L}{2}$ and its mass is $m = \rho V$, so separating $\rho$
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
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<meta name="description" content="The largest dataset of solutions of 'Savchenko. Problems in Physics'. Savchenko’s Problems in General Physics is widely used to prepare for olympiads and it is a useful tool to
<meta name="description" content="Two equal trunks are placed as the figure shows. The lower trunk is linked to the vertical wall by wires that form an angle of 45^\circ with it. The upper trunk is half submerged in water. Find the density of the trunks.">
master and sharpen your skills and techniques in comptetitive problem solving. Some of these problems were a source
of inspiration for Jaan Kalda’s handouts and to some NBPhO problems. You may find problems from old IPhO
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<meta property="og:description" content="Two equal trunks are placed as the figure shows. The lower trunk is linked to the vertical wall by wires that form an angle of 45^\circ with it. The upper trunk is half submerged in water. Find the density of the trunks.">
<title>Two equal trunks are placed as the figure shows. The lower trunk is linked to the vertical wall by wires that form an angle of 45^\circ with it. The upper trunk is half submerged in water. Find the density of the trunks.</title>
Two equal trunks are placed as the figure shows. The lower trunk is linked to the vertical wall by wires that form an angle of 45$^\circ$ with it. The upper trunk is half submerged in water. Find the density of the trunks.
Two equal trunks are placed as the figure shows. The lower trunk is linked to the vertical wall by wires that form an angle of 45$^\circ$ with it. The upper trunk is half submerged in water. Find the density of the trunks.
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="statement.png"
<img src="statement.png"
loading="lazy" alt="4.2.13" width="300" />
loading="lazy" alt="4.2.13" width="300" />
<figcaption>
<figcaption>
For problem 4.2.13
For problem 4.2.13
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
Let's analyze the following force diagram
Let's analyze the following force diagram
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="diagram.png"
<img src="diagram.png"
loading="lazy" alt="4.2.13" width="300" />
loading="lazy" alt="4.2.13" width="300" />
<figcaption>
<figcaption>
Force diagram for each trunk
Force diagram for each trunk
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
Consider the trunks are perfect wooden cylinders. Applying Newton Second Law for cylinder 1, on $x$-direction,
Consider the trunks are perfect wooden cylinders. Applying Newton Second Law for cylinder 1, on $x$-direction,
$$-N'\cos{45^\circ}+N = 0 \;(1)$$
$$-N'\cos{45^\circ}+N = 0 \;(1)$$
and on $y$-axis
and on $y$-axis
$$mg = \rho_0 g V_s + N'\sin{45^\circ} = \rho_0 g V_{s1} + N'\frac{\sqrt{2}}{2}$$
$$mg = \rho_0 g V_s + N'\sin{45^\circ} = \rho_0 g V_{s1} + N'\frac{\sqrt{2}}{2}$$
as trunk 1 is half submerged, $V_{s1} = \frac{\pi R^2 L}{2}$ and its mass is $m = \rho V$, so separating $\rho$
as trunk 1 is half submerged, $V_{s1} = \frac{\pi R^2 L}{2}$ and its mass is $m = \rho V$, so separating $\rho$
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
