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<meta name="description" content="A cube of side a, made of a material of density
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ho, is found on container's bottom that forms an angle lpha with the horizon. Find the force which the cube acts over bottom, if a liquid of density
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ho_0 fills the container. The superior side of cube is at distance h from the liquid's surface. Between the container's bottom and the cube there is no liquid. The atmospheric pressure is P and the acceleration due to gravity is g.">
<meta property="og:title" content="A cube of side a, made of a material of density
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ho, is found on container's bottom that forms an angle lpha with the horizon. Find the force which the cube acts over bottom, if a liquid of density
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ho_0 fills the container. The superior side of cube is at distance h from the liquid's surface. Between the container's bottom and the cube there is no liquid. The atmospheric pressure is P and the acceleration due to gravity is g.">
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<meta property="og:description" content="A cube of side a, made of a material of density
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ho, is found on container's bottom that forms an angle lpha with the horizon. Find the force which the cube acts over bottom, if a liquid of density
+
ho_0 fills the container. The superior side of cube is at distance h from the liquid's surface. Between the container's bottom and the cube there is no liquid. The atmospheric pressure is P and the acceleration due to gravity is g.">
<title>A cube of side a, made of a material of density
+
ho, is found on container's bottom that forms an angle lpha with the horizon. Find the force which the cube acts over bottom, if a liquid of density
+
ho_0 fills the container. The superior side of cube is at distance h from the liquid's surface. Between the container's bottom and the cube there is no liquid. The atmospheric pressure is P and the acceleration due to gravity is g.</title>
A cube of side $a$, made of a material of density $\rho$, is found on container's bottom that forms an angle $\alpha$ with the horizon. Find the force which the cube acts over bottom, if a liquid of density $\rho_0$ fills the container. The superior side of cube is at distance $h$ from the liquid's surface. Between the container's bottom and the cube there is no liquid. The atmospheric pressure is $P$ and the acceleration due to gravity is $g$.
</p>
<center>
<figure>
<img src="statement.png"
loading="lazy" alt="4.1.13" width="280" />
<figcaption>
For problem 4.1.13
</figcaption>
</figure>
</center>
<h3>Solution</h3>
<p>
Let's define the $y$-axis perpendicular to bottom and the $x$-axis parallel to it. At $y$-axis there is no buoyancy force because between bottom and the cube there is no liquid.
</p>
<center>
<figure>
<img src="draw1.png"
loading="lazy" alt="4.1.13" width="450" />
<figcaption>
Pressure analysis at $x$-axis
</figcaption>
</figure>
</center>
<p>
The pressures at points $A$ and $C$ are
$$P_A = P + \rho_0 g(h+a\sin{\alpha})$$
$$P_C = P + \rho_0 gh$$
If both equations are rested,
$$P_A-P_C = \rho ga\sin{\alpha} \;(1)$$
Note that difference of pressures between any point over side $AB$ and any point over $CD$ is expressed by $(1)$ (for any two points at same $y$ and at $a$ distance one from the other). If $(1)$ is multiplied by the side area $a^2$, buoyancy force can be found for $x$-axis
$$F_x = \rho g a^3 \sin{\alpha} \;(2)$$
Applying Newton's Second Law for $x$-axis, also considering friction force
$$mg\sin{\alpha} = f_r + F_x$$
As $m = \rho a^3$ and according $(2)$
$$\boxed{f_r = a^3 g (\rho-\rho_0)\sin{\alpha}}$$
This force is "felt" by bottom in opposed direction of $x$-axis. Now, pressure force $F_p$ which acts in $y$-axis, generated by above water mass and the atmospher, must be found.
</p>
<center>
<figure>
<img src="draw2.png"
loading="lazy" alt="4.1.13" width="480" />
<figcaption>
Pressure analysis at $y$-axis
</figcaption>
</figure>
</center>
<p>
Atmospheric-pressure force is
$$F_0 = \frac{Pa^2}{\cos{\alpha}}$$
but the cube doesn't "feel" that force, else a component of it ($F_{0y}$).
$$F_{0y} = F_0\cos{\alpha} = Pa^2 \;(3)$$
The pressure exerted by water mass is
$$F_w = m_w g \cos{\alpha} = \rho_0 V_w g \cos{\alpha} \;(4)$$
<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">
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<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="A cube of side a, made of a material of density
master and sharpen your skills and techniques in comptetitive problem solving. Some of these problems were a source
ho, is found on container's bottom that forms an angle lpha with the horizon. Find the force which the cube acts over bottom, if a liquid of density
of inspiration for Jaan Kalda’s handouts and to some NBPhO problems. You may find problems from old IPhO
ho_0 fills the container. The superior side of cube is at distance h from the liquid's surface. Between the container's bottom and the cube there is no liquid. The atmospheric pressure is P and the acceleration due to gravity is g.">
<meta property="og:title" content="A cube of side a, made of a material of density
ho, is found on container's bottom that forms an angle lpha with the horizon. Find the force which the cube acts over bottom, if a liquid of density
ho_0 fills the container. The superior side of cube is at distance h from the liquid's surface. Between the container's bottom and the cube there is no liquid. The atmospheric pressure is P and the acceleration due to gravity is g.">
<meta property="og:image" content="img/logo.png">
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<meta property="og:description" content="A website with solutions to physics problems from Savchenko Textbook">
<meta property="og:description" content="A cube of side a, made of a material of density
ho, is found on container's bottom that forms an angle lpha with the horizon. Find the force which the cube acts over bottom, if a liquid of density
ho_0 fills the container. The superior side of cube is at distance h from the liquid's surface. Between the container's bottom and the cube there is no liquid. The atmospheric pressure is P and the acceleration due to gravity is g.">
<title>A cube of side a, made of a material of density
ho, is found on container's bottom that forms an angle lpha with the horizon. Find the force which the cube acts over bottom, if a liquid of density
ho_0 fills the container. The superior side of cube is at distance h from the liquid's surface. Between the container's bottom and the cube there is no liquid. The atmospheric pressure is P and the acceleration due to gravity is g.</title>
A cube of side $a$, made of a material of density $\rho$, is found on container's bottom that forms an angle $\alpha$ with the horizon. Find the force which the cube acts over bottom, if a liquid of density $\rho_0$ fills the container. The superior side of cube is at distance $h$ from the liquid's surface. Between the container's bottom and the cube there is no liquid. The atmospheric pressure is $P$ and the acceleration due to gravity is $g$.
A cube of side $a$, made of a material of density $\rho$, is found on container's bottom that forms an angle $\alpha$ with the horizon. Find the force which the cube acts over bottom, if a liquid of density $\rho_0$ fills the container. The superior side of cube is at distance $h$ from the liquid's surface. Between the container's bottom and the cube there is no liquid. The atmospheric pressure is $P$ and the acceleration due to gravity is $g$.
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="statement.png"
<img src="statement.png"
loading="lazy" alt="4.1.13" width="280" />
loading="lazy" alt="4.1.13" width="280" />
<figcaption>
<figcaption>
For problem 4.1.13
For problem 4.1.13
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
Let's define the $y$-axis perpendicular to bottom and the $x$-axis parallel to it. At $y$-axis there is no buoyancy force because between bottom and the cube there is no liquid.
Let's define the $y$-axis perpendicular to bottom and the $x$-axis parallel to it. At $y$-axis there is no buoyancy force because between bottom and the cube there is no liquid.
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="draw1.png"
<img src="draw1.png"
loading="lazy" alt="4.1.13" width="450" />
loading="lazy" alt="4.1.13" width="450" />
<figcaption>
<figcaption>
Pressure analysis at $x$-axis
Pressure analysis at $x$-axis
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
The pressures at points $A$ and $C$ are
The pressures at points $A$ and $C$ are
$$P_A = P + \rho_0 g(h+a\sin{\alpha})$$
$$P_A = P + \rho_0 g(h+a\sin{\alpha})$$
$$P_C = P + \rho_0 gh$$
$$P_C = P + \rho_0 gh$$
If both equations are rested,
If both equations are rested,
$$P_A-P_C = \rho ga\sin{\alpha} \;(1)$$
$$P_A-P_C = \rho ga\sin{\alpha} \;(1)$$
Note that difference of pressures between any point over side $AB$ and any point over $CD$ is expressed by $(1)$ (for any two points at same $y$ and at $a$ distance one from the other). If $(1)$ is multiplied by the side area $a^2$, buoyancy force can be found for $x$-axis
Note that difference of pressures between any point over side $AB$ and any point over $CD$ is expressed by $(1)$ (for any two points at same $y$ and at $a$ distance one from the other). If $(1)$ is multiplied by the side area $a^2$, buoyancy force can be found for $x$-axis
$$F_x = \rho g a^3 \sin{\alpha} \;(2)$$
$$F_x = \rho g a^3 \sin{\alpha} \;(2)$$
Applying Newton's Second Law for $x$-axis, also considering friction force
Applying Newton's Second Law for $x$-axis, also considering friction force
$$mg\sin{\alpha} = f_r + F_x$$
$$mg\sin{\alpha} = f_r + F_x$$
As $m = \rho a^3$ and according $(2)$
As $m = \rho a^3$ and according $(2)$
$$\boxed{f_r = a^3 g (\rho-\rho_0)\sin{\alpha}}$$
$$\boxed{f_r = a^3 g (\rho-\rho_0)\sin{\alpha}}$$
This force is "felt" by bottom in opposed direction of $x$-axis. Now, pressure force $F_p$ which acts in $y$-axis, generated by above water mass and the atmospher, must be found.
This force is "felt" by bottom in opposed direction of $x$-axis. Now, pressure force $F_p$ which acts in $y$-axis, generated by above water mass and the atmospher, must be found.
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="draw2.png"
<img src="draw2.png"
loading="lazy" alt="4.1.13" width="480" />
loading="lazy" alt="4.1.13" width="480" />
<figcaption>
<figcaption>
Pressure analysis at $y$-axis
Pressure analysis at $y$-axis
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
Atmospheric-pressure force is
Atmospheric-pressure force is
$$F_0 = \frac{Pa^2}{\cos{\alpha}}$$
$$F_0 = \frac{Pa^2}{\cos{\alpha}}$$
but the cube doesn't "feel" that force, else a component of it ($F_{0y}$).
but the cube doesn't "feel" that force, else a component of it ($F_{0y}$).
$$F_{0y} = F_0\cos{\alpha} = Pa^2 \;(3)$$
$$F_{0y} = F_0\cos{\alpha} = Pa^2 \;(3)$$
The pressure exerted by water mass is
The pressure exerted by water mass is
$$F_w = m_w g \cos{\alpha} = \rho_0 V_w g \cos{\alpha} \;(4)$$
$$F_w = m_w g \cos{\alpha} = \rho_0 V_w g \cos{\alpha} \;(4)$$
<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>
