To solve the problem involving a non-conducting rod pivoted at its center with weights and charges, we need to analyze the forces and torques acting on the system. Let's break this down step by step.

Understanding the Setup

We have a non-conducting rod of length l that is pivoted at its center. This means that the pivot point divides the rod into two equal halves, each of length l/2. A weight w is placed at a distance X from the left end of the rod. Additionally, there are two small conducting spheres attached to the ends of the rod, one with charge q at the left end and another with charge 2q at the right end. The distance from the fixed charges Q to the spheres is r.

Analyzing Forces and Torques

To find the distance X, we need to consider the torques about the pivot point. The torque caused by a force is given by the formula:

• Torque (τ) = Force (F) × Distance (d)

In our case, we have two main forces acting on the rod:

• The weight w acting downwards at a distance X - l/2 from the pivot (since the pivot is at the center).
• The electrostatic forces due to the charges at the ends of the rod.

Calculating the Torque Due to Weight

The torque due to the weight w can be expressed as:

• τ_w = w × (X - l/2)

Calculating the Torque Due to Charges

Next, we need to consider the electrostatic forces acting on the rod due to the charges q and 2q. The force between two charges is given by Coulomb's law:

• F = k × (|q1 × q2|) / r²

Here, k is Coulomb's constant. The force acting on the charge q at the left end due to charge Q will create a torque about the pivot. Similarly, the charge 2q at the right end will also exert a force that creates torque.

Setting Up the Torque Balance Equation

For the system to be in equilibrium, the sum of the torques about the pivot must equal zero:

• τ_w + τ_q + τ_2q = 0

Substituting the expressions for torque, we have:

• w × (X - l/2) + τ_q + τ_2q = 0

Finding the Distances for Charges

The distances for the charges from the pivot are:

• For charge q: (-l/2) (to the left of the pivot)
• For charge 2q: (l/2) (to the right of the pivot)

Thus, the torques due to the charges can be calculated as:

• τ_q = F_q × (-l/2)
• τ_2q = F_2q × (l/2)

Solving for X

Now we can substitute the forces due to the charges into the torque balance equation and solve for X. This will involve substituting the expressions for the forces from Coulomb's law into the torque equations and simplifying. The final equation will allow us to isolate X.

In summary, the key steps involve calculating the torques due to the weight and the electrostatic forces, setting up the equilibrium condition, and solving for the unknown distance X. The exact numerical solution will depend on the specific values of w, q, 2q, r, and l.