The Figure(for figure please refer to h.c.verma part 2...page36) shows a cylindrical tube of cross sectional area A filled 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 when the tempearture is increased to 2T₀?

I am very lousy in Physics so...please show me how to solve it too.....not just the ans.....Thank You.

Let's break down the problem step by step to understand how to find the tension in the wire connecting the two pistons in the cylindrical tube. We will analyze the situation at two different temperatures: the initial temperature \( T_0 \) and the increased temperature \( 2T_0 \). This will help us see how the tension changes with temperature.

Understanding the Initial Conditions

Initially, we have a gas in a cylinder with a pressure \( p_0 \) equal to atmospheric pressure. The pistons are frictionless, which means they can move without any resistance. The tension in the wire connecting the pistons is influenced by the pressure of the gas acting on the pistons.

Step 1: Analyzing the Initial Tension

At the initial state, the pressure \( p_0 \) is acting on the pistons. The force exerted by the gas on each piston can be calculated using the formula:

• Force (F) = Pressure (p) × Area (A)

Since the pressure is \( p_0 \) and the area is \( A \), the force on each piston is:

• F = \( p_0 \times A \)

Now, since the two pistons are connected by a wire, the tension \( T \) in the wire must balance the forces acting on the pistons. Each piston experiences the same force due to the gas pressure, so the tension in the wire is equal to the force exerted by one piston:

• Tension (T) = \( p_0 \times A \)

Calculating the Tension at Initial Temperature

At the initial temperature \( T_0 \), the tension in the wire is:

• T = \( p_0 \times A \)

Step 2: Analyzing the Situation at Increased Temperature

Now, let's consider what happens when the temperature of the gas is increased to \( 2T_0 \). According to the ideal gas law, the pressure of a gas is directly proportional to its temperature when the volume is held constant. This can be expressed as:

• Pressure (p) ∝ Temperature (T)

Thus, if the temperature doubles from \( T_0 \) to \( 2T_0 \), the pressure will also double, assuming the volume remains constant. Therefore, the new pressure \( p \) at temperature \( 2T_0 \) can be expressed as:

• p = \( 2p_0 \)

Finding the New Tension

With the new pressure, we can calculate the new tension in the wire. The force exerted by the gas on each piston at the new pressure is:

• F = \( 2p_0 \times A \)

Since the tension in the wire must balance this force, the new tension \( T' \) is:

• T' = \( 2p_0 \times A \)

Summary of Results

To summarize, we have:

• Initial tension when temperature is \( T_0 \): T = \( p_0 \times A \)
• New tension when temperature is increased to \( 2T_0 \): T' = \( 2p_0 \times A \)

So, the tension in the wire doubles when the temperature of the gas is increased from \( T_0 \) to \( 2T_0 \). This illustrates the direct relationship between temperature and pressure in a gas, and how it affects the forces acting on the pistons. If you have any further questions or need clarification on any part, feel free to ask!