Question

A small ball with a mass of \(30.0 \mathrm{~g}\) and a charge of \(-0.200 \mu \mathrm{C}\) is suspended from the ceiling by a string. The ball hangs at a distance of \(5.00 \mathrm{~cm}\) above an insulating floor. If a second small ball with a mass of \(50.0 \mathrm{~g}\) and a charge of \(0.400 \mu \mathrm{C}\) is rolled directly beneath the first ball, will the second ball leave the floor? What is the tension in the string when the second ball is directly beneath the first ball?

Short answer

Answer: Yes, the second ball will leave the floor. The tension in the string is 0.2943 N.

Step-by-step solution

Determine the gravitational forces

First, we need to calculate the gravitational forces acting upon both balls. The gravitational force can be determined using the formula: \(F_g = mg\), where \(m\) is the mass and \(g\) is the acceleration due to gravity (approximately \(9.81 \mathrm{m/s^2}\)). For the first ball: \(F_{g1} = (30.0 \mathrm{g}) (9.81 \mathrm{m/s^2}) = 0.0300 \mathrm{kg} \times 9.81 \mathrm{m/s^2} = 0.2943 \mathrm{N}\) For the second ball: \(F_{g2} = (50.0 \mathrm{g}) (9.81 \mathrm{m/s^2}) = 0.0500 \mathrm{kg} \times 9.81 \mathrm{m/s^2} = 0.4905 \mathrm{N}\)

Find the electrostatic force between the balls

Next, we need to calculate the electrostatic force between the two balls using Coulomb's law. The formula for Coulomb's law is: \(F_e = K \frac{|q_1q_2|}{r^2}\), where \(K=8.99\times10^9\,\mathrm{N\cdot m^2/C^2}\), \(q_1\) and \(q_2\) are charges of the balls, and \(r\) is the distance between them. Given charges: \(q_1 = -0.200 \mathrm{\,\mu C}\) and \(q_2 = 0.400 \mathrm{\,\mu C}\) Distance between the balls: \(r = 5.00 \mathrm{cm} = 0.0500\,\mathrm{m}\) Now, placing the given values into the formula: \(F_e = 8.99 \times 10^9 \mathrm{\,N\cdot m^2/C^2} \frac{|(-0.2 \times 10^{-6} \mathrm C)(0.4 \times 10^{-6} \mathrm C)|}{(0.0500 \mathrm{m})^2} = 5.76 \mathrm{N}\)

Determine if the second ball will leave the floor

To see if the second ball will leave the floor, compare the gravitational force \(F_{g2}\) with the electrostatic force \(F_e\) acting on it. If the electrostatic force is greater, the second ball will leave the floor. Since \(F_e > F_{g2}\) (\(5.76 \mathrm{N} > 0.4905\,\mathrm{N}\)), the second ball will leave the floor.

Calculate the tension in the string

In equilibrium, the tension in the string \(T\) must balance the gravitational force acting on the first ball (\(F_{g1}\)) and the component of the electrostatic force (\(F_e\)) acting in the upward direction (as second ball repels the first ball). As the balls are directly below each other, and the angle between the electrostatic force and the string is \(90^\circ\), the two forces are perpendicular and can be treated independently in this particular case. Therefore, the tension in the string is equal to the gravitational force acting on the first ball: \(T = F_{g1} = 0.2943\,\mathrm{N}\).

Key concepts

Coulomb's Law

Understanding the interactions between charged objects is foundational in physics, and Coulomb's law provides a quantitative description of the electrostatic force between two point charges. The law is named after French physicist Charles-Augustin de Coulomb who introduced it in the 1780s. The formula for Coulomb's law is expressed as \(F_e = K \frac{|q_1q_2|}{r^2}\), where \(F_e\) is the electrostatic force, \(K\) is Coulomb's constant (\(8.99 \times 10^9 \,\mathrm{N\cdot m^2/C^2}\)), \(q_1\) and \(q_2\) are the magnitudes of the charges, and \(r\) is the distance between the centers of the two charges.

In layman's terms, the force increases as the charges increase or the distance between them decreases. The force is attractive if the charges are opposite, and repulsive if the charges are the same. This principle helps us analyze situations where charged objects influence each other, like in the exercise where one charged ball affects the position of another.

Gravitational Force

Every mass exerts an attractive gravitational force on every other mass. This is a key concept that governs the motion of objects from apples falling from trees to the movement of celestial bodies. The formula to calculate it on Earth is \(F_g = mg\), where \(m\) is the mass of the object and \(g\) is the acceleration due to gravity, approximately \(9.81 \,\mathrm{m/s^2}\).

Although gravity is the weakest of the fundamental forces, it is significant for larger masses and distances. This concept is demonstrated in the exercise where we calculate the force of gravity acting on both balls, which in turn helps us determine whether one can suspend the other.

Equilibrium in Physics

Equilibrium is a crucial concept in physics, indicating a state in which all forces acting upon an object are balanced, resulting in no change in motion. It is essential to solve problems involving stationary or moving objects where the net forces and net torques are zero.

Objects in equilibrium are either at rest (static equilibrium) or moving with a constant velocity (dynamic equilibrium). In the context of the exercise, the first ball is initially in static equilibrium, with the tension in the string balancing the gravitational force. When the second charged ball approaches, the electrostatic repulsive force alters this equilibrium, requiring a new calculation for the tension to maintain equilibrium.

Electric Charge

Electric charge is a fundamental property of particles that determines how they interact electromagnetically. There are two types of electric charges: positive and negative. Like charges repel each other, while unlike charges attract. The unit of charge is the Coulomb (\(C\)), but charges can also be measured in microcoulombs (\(\mu C\)), as seen in this exercise.

The concept of charge is not just a theoretical idea but underpins much of modern technology, from the smallest computer components to large-scale electrical grids. In the exercise, the electric charge is essential in calculating the electrostatic force exerted between the two balls, using Coulomb's law, thus playing a pivotal role in determining the mechanical equilibrium of the system.