Question

Figure shows a cylindrical tube of cross-sectional area A fitted with two frictionless pistons. The pistons are connected to each other by a metallic wire. Initially, the temperature of the gas is T0 and its pressure is p0 which equals the atmospheric pressure. (a) What is the tension in the wire? (b) What will be the tension if the temperature is increased to 2T0?
Figure

Answer 1

(a) Tension of the wire is zero

(b) Tension at temperature increased to 2$T_0$ is  $T = P_oA$

Explanation:

Given Data

Initial temperature = $T_0$

Initial Pressure = $P_0$

To find (a) Tension and (b) Tension if the temperature increased to 2$T_0$

(a) Since external and internal pressure of the cylinder is the same, the pistons are not subject to net pressure. So, tension is going to be zero.

(b) $T_1 = T_0$

$T_2 = 2T_o$

$P_1 = P_0 = 105 Pa$

Cross Sectional Area =A

Let the distance between the pistons be L.

Volume =Area × length

When we apply five variable gas equations, we get

$\frac{P_{2} V}{T_{1}}=\frac{P_{2} V}{T_{2}}$

$\frac{10^{5}}{T_{0}}=\frac{P_{2}}{2 T_{0}}$

$P_2=2 \times 105 = 2P_o$

Net force which acts outside = $2P_0 - P_0=P_0$

Force at a piston $F= P_o \times A$

We get by the free body diagram

F -T=0

$T = P_oA$

For a cylindrical tube with two pistons connected with metallic wire, with initial temperature and pressure be $T_0$ and $P_0$ the tension in the wire is zero due to same internal pressure of the cylinder and the tension when temperature increased to 2$T_0$  is  $P_oA$