To tackle this problem, we need to analyze the forces acting on the non-conducting rod and the charges attached to it. The setup involves a rod pivoted at its center, with a weight and charges positioned at specific distances. Our goal is to find the distance \( X \) from the left end of the rod where the weight \( w \) is placed. Let's break this down step by step.

Understanding the System

The rod is non-conducting and is pivoted at its center, meaning it can rotate around this pivot point. The weight \( w \) is applied at a distance \( X \) from the left end of the rod. At each end of the rod, we have small conducting spheres with charges \( q \) and \( 2q \), respectively, positioned at a distance \( r \) from fixed charges \( Q \). The forces acting on the rod include gravitational force due to the weight and electrostatic forces due to the charges.

Forces Acting on the Rod

• Gravitational Force: The weight \( w \) exerts a downward force at distance \( X \) from the left end.
• Electrostatic Forces: The charges \( q \) and \( 2q \) will experience forces due to the fixed charges \( Q \). These forces will depend on the distance \( r \) and the magnitudes of the charges.

Setting Up the Torque Equation

To find the equilibrium position of the rod, we need to set the net torque around the pivot point to zero. The torque \( \tau \) due to a force is given by the formula:

Torque = Force × Distance from pivot

Calculating Torque from the Weight

The torque due to the weight \( w \) is:

Torque from weight = w × (X - l/2)

Here, \( l/2 \) is the distance from the pivot to the left end of the rod.

Calculating Torque from the Charges

For the charges, we need to consider the forces acting on each charge due to the fixed charge \( Q \). The force \( F \) on a charge due to another charge is given by Coulomb's law:

F = k × (|q_1 × q_2|) / r^2

where \( k \) is Coulomb's constant. The forces on the charges \( q \) and \( 2q \) will create torques about the pivot point as well.

Equilibrium Condition

For the rod to be in equilibrium, the sum of the torques must equal zero:

Torque from weight + Torque from charge q + Torque from charge 2q = 0

Solving for X

Assuming the distances from the pivot to the charges are \( l/2 - r \) for charge \( q \) and \( l/2 + r \) for charge \( 2q \), we can express the torques due to these charges. The equilibrium condition can then be set up as:

w × (X - l/2) = F_q × (l/2 - r) + F_{2q} × (l/2 + r)

Substituting the expressions for \( F_q \) and \( F_{2q} \) from Coulomb's law will allow us to solve for \( X \). This will involve some algebraic manipulation to isolate \( X \).

Final Steps

After substituting and simplifying, you will arrive at an expression for \( X \) in terms of the known quantities \( w \), \( q \), \( 2q \), \( r \), and \( Q \). This will give you the distance from the left end of the rod where the weight should be placed to maintain equilibrium.

In summary, the key to solving this problem lies in understanding the forces and torques acting on the system and applying the principle of equilibrium. By carefully setting up the equations and solving for \( X \), you can find the required distance for the weight to balance the system effectively.