<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="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.">
<meta property="og:title" content="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.">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<meta property="og:description" content="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.">
<title>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.</title>
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>
<h3>Solution</h3>
<p>
Let denote left volume as $V_1$ and right one as $V_2$. Applying state equation for each container
$$P_1~V_1 = nRT \;(1)$$
$$P_0~V_2 = nRT_0 \;(2)$$
It is easy to see that
$$V_1 = V_0+S\frac{l}{2}-Sx \;(3)$$
$$V_2 = V_0+S\frac{l}{2}+Sx \;(4)$$
If droplet is in equilibrium that means pressure forces due to gasses are equal, so $P_1 = P_0$. Then, dividing $(1)$ and $(2)$
$$\frac{V_1}{V_2}=\frac{T}{T_0} \;(5)$$
Substituting $(3)$ and $(4)$ into $(5)$ and separating $T$
<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>
<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 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.">
<meta name="description" content="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.">
<meta property="og:title" content="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.">
<meta property="og:title" content="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.">
<meta property="og:image" content="img/logo.png">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<meta property="og:description" content="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.">
<meta property="og:description" content="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.">
<title>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.</title>
<title>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.</title>
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.
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>
</p>
<center>
<center>
<figure>
<figure>
<img src="statement.png"
<img src="statement.png"
loading="lazy" alt="5.5.16" width="250" />
loading="lazy" alt="5.5.16" width="250" />
<figcaption>
<figcaption>
For problem 5.5.16
For problem 5.5.16
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
Let denote left volume as $V_1$ and right one as $V_2$. Applying state equation for each container
Let denote left volume as $V_1$ and right one as $V_2$. Applying state equation for each container
$$P_1~V_1 = nRT \;(1)$$
$$P_1~V_1 = nRT \;(1)$$
$$P_0~V_2 = nRT_0 \;(2)$$
$$P_0~V_2 = nRT_0 \;(2)$$
It is easy to see that
It is easy to see that
$$V_1 = V_0+S\frac{l}{2}-Sx \;(3)$$
$$V_1 = V_0+S\frac{l}{2}-Sx \;(3)$$
$$V_2 = V_0+S\frac{l}{2}+Sx \;(4)$$
$$V_2 = V_0+S\frac{l}{2}+Sx \;(4)$$
If droplet is in equilibrium that means pressure forces due to gasses are equal, so $P_1 = P_0$. Then, dividing $(1)$ and $(2)$
If droplet is in equilibrium that means pressure forces due to gasses are equal, so $P_1 = P_0$. Then, dividing $(1)$ and $(2)$
$$\frac{V_1}{V_2}=\frac{T}{T_0} \;(5)$$
$$\frac{V_1}{V_2}=\frac{T}{T_0} \;(5)$$
Substituting $(3)$ and $(4)$ into $(5)$ and separating $T$
Substituting $(3)$ and $(4)$ into $(5)$ and separating $T$
<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>
<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>
