Chapter 21: Problem 53
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 \(\mathrm{a}\) 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
Expert verified
Answer: Yes, the second ball will leave the floor when it is directly beneath the first ball since the electrostatic force is greater than the gravitational force acting on it. The tension in the string when the second ball is directly beneath the first ball is 0.87 N.
Step by step solution
01
List down given information and convert units
First, we need to list down the given information and convert the units to SI units.
- Mass of the first ball, \(m_1 = 30.0\,\text{g} = 0.030\,\text{kg}\)
- Charge of the first ball, \(q_1 = -0.200\,\mu\text{C} = -2.00\times 10^{-7}\,\text{C}\)
- Mass of the second ball, \(m_2 = 50.0\,\text{g} = 0.050\,\text{kg}\)
- Charge of the second ball, \(q_2 = 0.400\,\mu\text{C} = 4.00\times 10^{-7}\,\text{C}\)
- Distance between the balls, \(r = 5.00\,\text{cm} = 0.050\,\text{m}\)
02
Calculate the electrostatic force between the balls
To calculate the electrostatic force between the balls, we will use Coulomb's law:
$$F_e = k\,\frac{q_1q_2}{r^2}$$
where \(k = 8.99\times 10^9\,\text{N}\cdot\text{m}^2\text{/C}^2\) is the electrostatic constant.
Plugging in the values, we get:
$$F_e = 8.99\times 10^9\,\frac{-2.00\times 10^{-7}\,\text{C}\cdot 4.00\times 10^{-7}\,\text{C}}{(0.050\,\text{m})^2}$$
So, the electrostatic force between the balls is:
$$F_e = -0.57\,\text{N}$$
The negative sign indicates that the force is attractive.
03
Calculate the gravitational force on the second ball
Now we will calculate the gravitational force acting on the second ball:
$$F_g = m_2g$$
where \(g = 9.81\,\text{m/s}^2\) is the acceleration due to gravity
Plugging in the values, we get:
$$F_g = 0.050\,\text{kg} \cdot 9.81\,\text{m/s}^2 $$
So, the gravitational force on the second ball is:
$$F_g = 0.49\,\text{N}$$
04
Determine if the second ball will leave the floor
To determine if the second ball will leave the floor, we need to compare the absolute values of the electrostatic and gravitational forces acting on it:
1. If \(|F_e| > F_g\), the second ball will leave the floor.
2. If \(|F_e| = F_g\), the second ball will hover but not leave the floor.
3. If \(|F_e| < F_g\), the second ball will stay on the floor.
In this case, since \(|-0.57\,\text{N}| > 0.49\,\text{N}\), the second ball will leave the floor.
05
Calculate the tension in the string when the second ball is directly beneath the first ball
When the second ball is directly beneath the first ball, the first ball is in equilibrium, so the net force acting on it is zero. Since the only forces acting on the first ball are the electrostatic force (vertically downward), gravitational force (also vertically downward), and the tension force (vertically upward), we can write:
$$T - F_{e1} - F_{g1} = 0$$
Here, \(F_{e1}\) is the electrostatic force acting on the first ball due to the second ball, \(F_{g1}=m_1g\) is the gravitational force acting on the first ball. Since the electrostatic force is attractive, we have \(F_{e1} = -F_e = 0.57\,\text{N}\).
Now, we can calculate the tension in the string:
$$T = F_{e1} + F_{g1} = 0.57\,\text{N} + (0.030\,\text{kg}\cdot 9.81\,\text{m/s}^2)$$
So, the tension in the string when the second ball is directly beneath the first ball is:
$$T = 0.87\,\text{N}$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coulomb's Law
To comprehend the interactions between charged objects, we employ Coulomb's law. This foundational principle quantifies the electrostatic force between two point charges. The formula is expressed as:
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Gravitational Force
Every object with mass exerts a gravitational force on every other object with mass. This attraction is described by Newton's law of universal gravitation and is given by the equation:
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Equilibrium of Forces
The concept of forces in equilibrium is essential in understanding how objects behave when subjected to multiple forces. In equilibrium, an object is either at rest or moving with constant velocity, indicating that the resultant force acting on it is zero. This state conforms to Newton's first law of motion.
In the scenario described in the exercise, the first ball remains in equilibrium when the second ball is placed beneath it. This means that the sum of forces acting on the first ball is zero – the upward tension force exerted by the string is balanced by the sum of the downward gravitational force and the attractive electrostatic force between the two charged balls.
To achieve equilibrium:\[\begin{equation}T = F_g + |F_e|\bigskip\bigskip\bigskip\begin{itemize}\bigskip\begin{itemize}\bigskip \begin{itemize}\bigskip \begin{itemize}\bigskip where \(T\) is the tension in the string, \(F_g\) is the gravitational force, and \(|F_e|\) is the magnitude of the electrostatic force. Understanding this balance of forces is crucial in many fields, such as engineering and physics, for analyzing the stability of structures or predicting the motion of objects under various forces.
In the scenario described in the exercise, the first ball remains in equilibrium when the second ball is placed beneath it. This means that the sum of forces acting on the first ball is zero – the upward tension force exerted by the string is balanced by the sum of the downward gravitational force and the attractive electrostatic force between the two charged balls.
To achieve equilibrium:\[\begin{equation}T = F_g + |F_e|\bigskip\bigskip\bigskip\begin{itemize}\bigskip\begin{itemize}\bigskip \begin{itemize}\bigskip \begin{itemize}\bigskip where \(T\) is the tension in the string, \(F_g\) is the gravitational force, and \(|F_e|\) is the magnitude of the electrostatic force. Understanding this balance of forces is crucial in many fields, such as engineering and physics, for analyzing the stability of structures or predicting the motion of objects under various forces.
