Two identical pith balls are charged by rubbing against each...

Two identical pith balls are charged by rubbing against each other. They are suspended from a horizontal rod through two strings of length $20 \mathrm{~cm}$ each, the separation between the suspension points being $5 \mathrm{~cm}$. In equilibrium, the separation between the balls is $3 \mathrm{~cm}$. Find the mass of each ball and the tension in the strings. The charge on each ball has a magnitude $2 \cdot 0 \times 10^{-8} \mathrm{C}$.

Key Concepts

Force Decomposition (Component Analysis)

Force decomposition involves breaking a force vector into its horizontal and vertical components. In this scenario, the tension in the strings is resolved into components such that the vertical component counteracts the gravitational force, and the horizontal component balances the electrostatic force between the balls. This analysis provides a pathway to set up the equations for equilibrium.

Geometrical Relationships in Pendulum Systems

The geometry of the pendulum system is important for establishing the angles at which the strings are inclined relative to the vertical. These angles are determined by the known lengths and distances in the setup, and they are essential for relating the tension components to the gravitational and electrostatic forces. Understanding the spatial arrangement allows for the correct application of trigonometric relationships in problem solving.

Electrostatic Forces (Coulomb's Law)

This concept involves the calculation of the force between two charged objects using Coulomb's law, which states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. It is crucial when determining the repulsive force that arises between the identically charged pith balls.

Equilibrium Conditions in Statics

In a system in static equilibrium, the net force acting on any object must be zero. This principle is applied both in the vertical and horizontal directions for the suspended balls, ensuring that the gravitational force is balanced by the vertical component of the string tension, and the horizontal electrostatic repulsion is balanced by the horizontal component of the tension.

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