To analyze the scenario you've described, we need to consider the forces at play between the two charged spheres and how the insulating rod affects the system. Let's break it down step by step.

Understanding the Setup

We have two small insulating spheres, each with a mass of 0.30 milligrams (or 0.00030 grams) and carrying equal but opposite charges of 15 picoCoulombs (pC). They are connected by a light, rigid insulating rod that is 5.0 cm long. Since the spheres are charged, they will exert electrostatic forces on each other.

Electrostatic Force Calculation

First, we can calculate the electrostatic force between the two spheres using Coulomb's Law, which is given by the formula:

F = k * |q1 * q2| / r²

Where:

• F is the force between the charges.
• k is Coulomb's constant, approximately 8.99 x 10^9 N m²/C².
• q1 and q2 are the magnitudes of the charges (both are 15 pC).
• r is the distance between the centers of the two spheres, which is equal to the length of the rod (5.0 cm or 0.05 m).

Substituting the values into the formula:

F = (8.99 x 10^9 N m²/C²) * (15 x 10^-12 C) * (15 x 10^-12 C) / (0.05 m)²

Calculating this gives:

F ≈ 0.809 N

Analyzing the Forces

This force of approximately 0.809 N is the electrostatic repulsion between the two spheres due to their like charges. Since they are connected by an insulating rod, they cannot move towards each other, but they will exert this force on each other. The rod will remain straight as long as the force does not exceed its tensile strength.

Effects of Mass and Gravity

Next, we should consider the effect of gravity on the spheres. The weight of each sphere can be calculated using the formula:

w = m * g

Where:

• w is the weight.
• m is the mass (0.30 mg = 0.00030 kg).
• g is the acceleration due to gravity (approximately 9.81 m/s²).

Calculating the weight:

w = 0.00030 kg * 9.81 m/s² ≈ 0.002943 N

Comparing Forces

Now, we can compare the electrostatic force and the gravitational force acting on each sphere. The electrostatic force (0.809 N) is significantly greater than the gravitational force (0.002943 N). This means that the repulsive force between the spheres will dominate the system's behavior.

Conclusion on Stability

In this setup, the spheres will experience a strong repulsive force due to their charges, which will keep them apart. The insulating rod will maintain their distance, and the system will remain stable as long as the rod can withstand the forces without breaking. If the rod were to fail, the spheres would move apart rapidly due to the electrostatic repulsion.

In summary, the interaction between the charged spheres is primarily governed by the electrostatic force, which far exceeds the gravitational force acting on them. This analysis highlights the importance of understanding both electrostatic and gravitational forces in systems involving charged objects.