Added Luis's English solution of 5.5.16

Luisito edited
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− <h2>Contents</h2>
− <ol>
− <li>
− <a href="#1" >Kinematics</a>
− <ol>
− <li><a href="#1.1">Motion with constant velocity</a></li>
− <li><a href="#1.2">Variable speed movement</a></li>
− <li><a href="#1.3">Motion in the field of gravity. Curvilinear motion</a></li>
− <li><a href="#1.4">Galileo's transformation</a></li>
− <li><a href="#1.5">Motion with links</a></li>
− </ol>
− </li>
− <li>
− <a href="#2">Dynamics</a>
− <ol>
− <li><a href="#2.1">Newton's laws</a></li>
− <li><a href="#2.2">Impulse. Center of mass</a></li>
− <li><a href="#2.3">Kinetic energy. Work. Potential energy</a></li>
− <li><a href="#2.4">Energy of the system. Energy transfer. Power</a></li>
− <li><a href="#2.5">Collisions</a></li>
− <li><a href="#2.6">Gravitational force. Kepler's laws</a></li>
− <li><a href="#2.7">Rotation of a rigid body</a></li>
− <li><a href="#2.8">Statics</a></li>
− </ol>
− </li>
− <li>
− <a href="#3">Oscillations and Waves</a>
− <ol>
− <li><a href="#3.1">Small deviations from equilibrium</a></li>
− <li><a href="#3.2">Period and frequency of free oscillations</a></li>
− <li><a href="#3.3">Harmonic motion</a></li>
− <li><a href="#3.4">Superposition of oscillations</a></li>
− <li><a href="#3.5">Forced and damped oscillations</a></li>
− <li><a href="#3.6">Deformations and stresses. Wave speed</a></li>
− <li><a href="#3.7">Wave propagation</a></li>
− <li><a href="#3.8">Superposition and reflection of waves</a></li>
− <li><a href="#3.9">Sound. Acoustic resonators</a></li>
− </ol>
− </li>
− <li>
− <a href="#4">Fluid Mechanics</a>
− <ol>
− <li><a href="#4.1">Pressure of a fluid</a></li>
− <li><a href="#4.2">Buoyancy. Archimedes' principle</a></li>
− <li><a href="#4.3">Motion of an ideal fluid</a></li>
− <li><a href="#4.4">Flow of a viscous fluid</a></li>
− <li><a href="#4.5">Surface tension of a liquid</a></li>
− <li><a href="#4.6">Capillary phenomena</a></li>
− </ol>
− </li>
− <li>
− <a href="#5">Molecular Physics</a>
− <ol>
− <li><a href="#5.1">Thermal motion of particles</a></li>
− <li><a href="#5.2">Distribution of gas molecules by speed</a></li>
− <li><a href="#5.3">Molecular collisions. Transport processes</a></li>
− <li><a href="#5.4">Rarefied gases. Interaction of molecules with the surface of a solid</a></li>
− <li><a href="#5.5">Equation of state of an ideal gas</a></li>
− <li><a href="#5.6">First law of thermodynamics. Heat capacity</a></li>
− <li><a href="#5.7">Gas effusion</a></li>
− <li><a href="#5.8">Probability of a thermodynamic state</a></li>
− <li><a href="#5.9">Second law of thermodynamics</a></li>
− <li><a href="#5.10">Phase transitions</a></li>
− <li><a href="#5.11">Thermal radiation</a></li>
− </ol>
− </li>
− <li>
− <a href="#6">Electrostatics</a>
− <ol>
− <li><a href="#6.1">Coulomb's law. Electric field intensity</a></li>
− <li><a href="#6.2">Flux of electric field intensity. Gauss's theorem</a></li>
− <li><a href="#6.3">Electric potential. Conductors in a steady electric field</a></li>
− <li><a href="#6.4">Capacitors</a></li>
− <li><a href="#6.5">Electric pressure. Energy of the electric field</a></li>
− <li><a href="#6.6">Electric field with dielectrics</a></li>
− </ol>
− </li>
− <li>
− <a href="#7">Motion of charged particles in an electric field</a>
− <ol>
− <li><a href="#7.1">Motion in a steady electric field</a></li>
− <li><a href="#7.2">Focusing of charged particles</a></li>
− <li><a href="#7.3">Motion in a variable electric field</a></li>
− <li><a href="#7.4">Interaction of charged particles</a></li>
− </ol>
− </li>
− <li>
− <a href="#8">Electric current</a>
− <ol>
− <li><a href="#8.1">Current. Current density. Current in a vacuum</a></li>
− <li><a href="#8.2">Conductance. Resistance. Sources of EMF</a></li>
− <li><a href="#8.3">Electrical circuits</a></li>
− <li><a href="#8.4">Capacitors and nonlinear elements in electrical circuits</a></li>
− </ol>
− </li>
− <li>
− <a href="#9">Steady magnetic field</a>
− <ol>
− <li><a href="#9.1">Magnetic field induction. Effect of magnetic field on current</a></li>
− <li><a href="#9.2">Magnetic field of a moving charge. Induction of a linear current</a></li>
− <li><a href="#9.3">Magnetic field of current distributed over surface or space</a></li>
− <li><a href="#9.4">Magnetic flux</a></li>
− </ol>
− </li>
− <li>
− <a href="#10">Motion of charged particles in complex fields</a>
− <ol>
− <li><a href="#10.1">Motion in a uniform magnetic field</a></li>
− <li><a href="#10.2">Drift motion of particles</a></li>
− </ol>
− </li>
− <li>
− <a href="#11">Electromagnetic induction</a>
− <ol>
− <li><a href="#11.1">Motion of conductors in a steady magnetic field. Electric motors</a></li>
− <li><a href="#11.2">Eddy electric field</a></li>
− <li><a href="#11.3">Mutual inductance. Inductance of conductors. Transformers</a></li>
− <li><a href="#11.4">AC electrical circuits</a></li>
− <li><a href="#11.5">Conservation of magnetic flux. Superconductors in a magnetic field</a></li>
− <li><a href="#11.6">Connection between variable electric and magnetic fields</a></li>
− </ol>
− </li>
− <li>
− <a href="#12">Electromagnetic waves</a>
− <ol>
− <li><a href="#12.1">Properties, emission, and reflection of electromagnetic waves</a></li>
− <li><a href="#12.2">Propagation of electromagnetic waves</a></li>
− </ol>
− </li>
− <li>
− <a href="#13">Geometrical optics. Photometry. Quantum nature of light</a>
− <ol>
− <li><a href="#13.1">Rectilinear propagation and reflection of light</a></li>
− <li><a href="#13.2">Refraction of light. Lens formula</a></li>
− <li><a href="#13.3">Optical systems</a></li>
− <li><a href="#13.4">Photometry</a></li>
− <li><a href="#13.5">Quantum nature of light</a></li>
− </ol>
− </li>
− <li>
− <a href="#14">Special theory of relativity</a>
− <ol>
− <li><a href="#14.1">Constancy of the speed of light. Addition of velocities</a></li>
− <li><a href="#14.2">Time dilation, length contraction in moving systems. Lorentz transformation</a></li>
− <li><a href="#14.3">Transformation of electric and magnetic fields</a></li>
− <li><a href="#14.4">Motion of relativistic particles in electric and magnetic fields</a></li>
− <li><a href="#14.5">Law of conservation of mass and momentum</a></li>
− </ol>
− </li>
− </ol>
−
−
<main>
<article>
− <h2 id="1" style="text-align: center;">Chapter 1. Kinematics</h2>
− <h3 id="1.1" style="text-align: center;">§ 1.1.Motion with constant velocity</h3>
− <div class="columns">
− <ul class="column">
− <li><a href="./1.1.1">1.1.1</a></li>
− <li><a href="./1.1.1">1.1.2</a></li>
− </ul>
− </div>
− <h3 id="1.2" style="text-align: center;">§ 1.2.Variable speed movement</h3>
− <h3 id="1.3" style="text-align: center;">§ 1.3.Motion in the field of gravity. Curvilinear motion</h3>
− <h3 id="1.4" style="text-align: center;">§ 1.4.Galileo's transformation</h3>
− <h3 id="1.5" style="text-align: center;">§ 1.5.Motion with links</h3>
−
+ <h3 id="back-link"><a href="../">$\leftarrow$Back</a></h3>
+
+ <h4>Statement</h4>
+ <p>
+ $5.5.16$
+ The gas thermometer consists in two identical containers with volume $V_0$ each one, linked by a tube of length $l$ and cross section $S$. A mercury droplet obstructs the tube. If temperatures of gasses in the volumes are equal, the mercury lies in the center of the tube. The right volume is connected to a thermostat with temperature $T_0$. Scale the thermometer finding the dependence between gas temperature in left volume and the displacement $x$ of mercury respect to equilibrium position.
+ </p>
+ <center>
+ <figure>
+ <img src="statement.png"
+ loading="lazy" alt="5.5.16" width="250" />
+ <figcaption>
+ For problem 5.5.16
+ </figcaption>
+ </figure>
+ </center>
+ <h4>Solution</h4>
+ <p class="TxtSolutions">
+ Let denote left volume as $V_1$ and right one as $V_2$. Applying state equation for each container
+ </p>
+ <center>
+ $P_1~V_1 = nRT \;(1)$
+ </center>
+ <center>
+ $P_0~V_2 = nRT_0 \;(2)$
+ </center>
+ <p class="TxtSolutions">
+ It is easy to see that
+ </p>
+ <center>
+ $V_1 = V_0+S\frac{l}{2}-Sx \;(3)$
+ </center>
+ <center>
+ $V_2 = V_0+S\frac{l}{2}+Sx \;(4)$
+ </center>
+ <p class="TxtSolutions">
+ If droplet is in equilibrium that means pressure forces due to gasses are equal, so $P_1 = P_0$. Then, dividing $(1)$ and $(2)$
+ </p>
+ <center>
+ \frac{V_1}{V_2}=\frac{T}{T_0} \;(5)$
+ </center>
+ Substituting $(3)$ and $(4)$ into $(5)$ and separating $T$
+ <center>
+ $\boxed{T = T_0~\frac{2V_0+S(l+2x)}{2V_0+S(l-2x)}}$
+ </center>
+ <p class="TxtSolutions">
+ This is mathematically valid for $0< x <\frac{V_0}{S}+\frac{l}{2}$ but according the physical situation only for $0< x < \frac{l}{2}$.
+ </p>
+ <p class="TxtSolutions" 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>
</article>
</main>
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