Question

Two balls have the same mass and the same charge, \(15.71 \mu \mathrm{C}\). They hang from the ceiling on strings of identical length, \(\ell=1.223 \mathrm{~m},\) as shown in the figure. The angle of the strings with respect to the vertical is \(21.07^{\circ} .\) What is the mass of each ball?

Short answer

Answer: The mass of each ball is approximately 0.107 kg.

Step-by-step solution

Identify the forces acting on the balls

The balls are hanging from the ceiling and experiencing gravitational force, which acts vertically downward, and the electrostatic repulsion force between them, which acts horizontally.

Calculate the electrostatic force between the balls

According to Coulomb's law, the electrostatic force between two charged objects is given by: \(F = k\frac{Q_1 Q_2}{r^2}\) where \(F\) is the electrostatic force, \(k\) is the electrostatic constant (\(8.99\times10^9 \mathrm{N m^2/C^2}\)), \(Q_1\) and \(Q_2\) are the charges on the two objects, and \(r\) is the distance between the centers of the objects. Let the distance between the centers of the balls be \(d\). Since the strings are of equal length (\(\ell = 1.223 \, \mathrm{m}\)) and the angle with respect to the vertical is the same (\(21.07^{\circ}\)) for both, we have an isosceles triangle formed by the strings and the horizontal line segment connecting the centers of the balls. Using trigonometry, we can solve for \(d\): \(d = 2\ell\sin\frac{\theta}{2}\) where \(\theta = 2 \times 21.07^{\circ} = 42.14^{\circ}\). \(d = 2(1.223)\sin(21.07)= 0.899 \, \mathrm{m}\) Now we can calculate the electrostatic force: \(F = k\frac{Q_1 Q_2}{d^2} = (8.99\times10^9)\frac{(15.71 \times 10^{-6})^2}{(0.899)^2} \approx 2.17895 \, \mathrm{N}\)

Analyze the forces acting on the balls

Now, let's consider one of the balls. Since the ball is in equilibrium (not accelerating), we can write Newton's second law for the forces acting on it: \(T\sin(\alpha) = F_e\) \(T\cos(\alpha) = mg\) where \(T\) is the tension in the string, \(\alpha\) is the angle of the string with respect to the vertical (\(21.07^{\circ}\)), \(F_e\) is the electrostatic force we calculated, and \(mg\) is the weight of the ball (with \(m\) being the mass we need to calculate, and \(g\) the acceleration due to gravity, approximately \(9.81 \, \mathrm{m/s^2}\)).

Solve for the mass

We can solve for the tension, \(T\), by dividing the two Newton's second law equations: \(\frac{T\sin(\alpha)}{T\cos(\alpha)} = \frac{F_e}{mg}\) \(T\) gets cancelled out, and we are left with: \(\tan(\alpha) = \frac{F_e}{mg}\) Now we can rearrange the formula to solve for \(m\): \(m = \frac{F_e}{g\tan(\alpha)} = \frac{2.17895}{(9.81)(\tan(21.07))} \approx 0.107 \, \mathrm{kg}\) So, the mass of each ball is approximately \(0.107 \, \mathrm{kg}\).

Key concepts

Understanding Electrostatic Force

Electrostatic force is a fundamental interaction between objects that have an electric charge. When we say that two charges are exerting an electrostatic force on each other, we are referring to the force described by Coulomb's law. This law states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between their centers.

In the exercise, two balls with identical charges are repelling each other due to this force. They are set up in such a way that their strings form equal angles on either side, revealing symmetry in the system. This setup allows us to determine the distance between the charges using basic trigonometry, which then permits the calculation of the electrostatic force acting on each ball.

Key Takeaways

• The electrostatic force is always present between objects with electric charge.
• Coulomb's law allows us to quantify this force.

Equilibrium of Forces

The equilibrium of forces occurs when the sum of all forces acting on an object is zero, and therefore, there is no net force and no acceleration. In our example with the charged balls, each ball is in equilibrium; the forces acting on it namely the tension in the string (both vertical and horizontal components) and the electrostatic force are balanced by gravity and the repulsion from the opposite charge.

To solve for the mass of the balls, we make use of this equilibrium condition. Carefully analyzing the vertical and horizontal components of the tension reveals the relationship between the electrostatic force and the gravitational force on the ball, leading us to solve for the mass.

Important Concepts

• In a static equilibrium, forces are balanced in all directions.
• Newton's second law can be adapted to systems in equilibrium to find unknown quantities.

Applying Trigonometry in Physics

The role of trigonometry in physics is crucial when dealing with problems involving angles and distances, such as our hanging charged balls. In this exercise, to determine the distance between the two balls, we used the sine function:
\[ d = 2\ell\sin\left(\frac{\theta}{2}\right) \]where \( d \) is the distance between the balls, \( \ell \) is the length of each string, and \( \theta \) is the total angle formed by the strings.

This trigonometric approach, along with the concept of symmetry in an isosceles triangle, gives us the necessary tools to find the horizontal separation between the charged balls, a critical step for applying Coulomb's law. Understanding how trigonometry shapes our calculations in physics is indispensable for solving many real-world problems.

Trigonometry Essentials

• Being comfortable with sine, cosine, and tangent functions is vital in physics.
• Trigonometry is used to relate angles with distances and resolve forces into components.