Question

The figure shows a horizontal cylinder with two sections of unequal cross-section in which two points of area $A_1$ and $A_2$ can move freely. The pistons are joined together with an inextensible string and the space between the pistons is filled with an ideal gas. The whole system is in equilibrium.

• A. the pressure of the gas is $2p_0$
• B. the pressure of the gas is $p_0$
• C. tension in the string is zero
• D. if the gas is heated, the pistons move towards right

Answer 1

Answer: (B), (C), (D)

The correct options are
B the pressure of the gas is $p_0$
C tension in the string is zero
D if the gas is heated, the pistons move towards right

Let $p$ be the pressure of the gas and $F$ be the tension in the string.
Considering the equilibrium of each piston,
For piston $1$:
$p{A}_1=p_0{A}_1+F\Rightarrow p=p_0+\frac{F}{A_1}$
...(Eq 1)
For piston 2:
$p{A}_2=p_0{A}_2+F$
$p=p_0+\frac{F}{A_2}$ ...(Eq 2)

From equations 1 and 2, if $A_1\ne A_2$ then the gas may have unique value of pressure only if tension in the string is zero, i.e., $F=0$ .
Thus, pressure of the gas is also atmospheric, $p=p_0$.
When the gas is heated, its volume increases at constant pressure, therefore, both the pistons shift towards right to accomodate the increased volume of the gas.