Two metal balls of mass \(m_{1}=5.00 \mathrm{~g}\) (diameter = \(5.00 \mathrm{~mm}\) ) and \(m_{2}=8.00 \mathrm{~g}\) (diameter \(=8.00 \mathrm{~mm}\) ) have positive charges of \(q_{1}=5.00 \mathrm{nC}\) and \(q_{2}=8.00 \mathrm{nC},\) respectively. A force holds them in place so that their centers are separated by \(8.00 \mathrm{~mm}\). What will their velocities be after the force is removed and they are separated by a large distance?

When considering phenomena like two charged metal balls in the absence of external forces, the principle of conservation of mechanical energy becomes applicable. Mechanical energy can be categorized into two forms: kinetic energy (KE), which is energy due to motion, and potential energy (PE), the stored energy based on position or configuration.

The conservation of mechanical energy states that in a closed system with no external work being done, the total mechanical energy remains constant. This concept is evident in our problem, where we initially have only electric potential energy due to the electrostatic force between the charges. As the balls are released and move apart, we expect their velocities to increase. This increase in kinetic energy must come from a decrease in potential energy since there is no external force doing work on the system.

By setting the initial potential energy equal to the sum of kinetic energies when the balls are far apart, we see energy transform from one form to the other while the total remains unchanged. This concept helps us to solve for the final velocities of the balls after enough time has passed that they are considered to be infinitely far from one another.

Another fundamental principle at play in our textbook example is the conservation of linear momentum, which states that in a system isolated from external forces, the total momentum remains constant over time.

Initially, both metal balls are at rest, which means the system has a total momentum of zero. Since no external forces act on the system when the balls are released, their momenta in opposite directions must cancel out to maintain that initial momentum of zero. Essentially, this means that the product of mass and velocity for one ball will be equal in magnitude and opposite in direction to the product for the other ball.

Using the linear momentum conservation, along with the information about the balls' masses, we deduce a relationship between their velocities. This relationship propels us towards resolving the unknown velocities by connecting them mathematically, thus allowing us to apply one equation (energy conservation) to solve for one variable at a time.

The interaction between two charged objects, like our metal balls, is governed by electrostatic force, a fundamental force arising from their electric charges. Coulomb's law quantitatively describes this force as directly proportional to the product of the charges and inversely proportional to the square of the distance between their centers.

In our scenario, the electric potential energy, which is effectively the stored energy of the system due to the electrostatic force, is given by the initial separation distance and the quantity of charge on each ball. When the electrostatic force is the only force doing work (as external forces are being held back), this potential energy will eventually convert completely into kinetic energy. Understanding the electrostatic force and how it affects potential energy is crucial for predicting the system's behavior once the force holding the balls together is removed - allowing the conservation of energy principle to take full effect and enabling us to calculate the final velocities.