Problem 63 - University Physics with Modern Physics

Question

Two small spheres with mass \(m =\) 15.0 g are hung by silk threads of length \(L =\) 1.20 m from a common point (Fig. P21.62). When the spheres are given equal quantities of negative charge, so that \(q_1 = q_2 = q\), each thread hangs at \(\theta = 25.0^\circ\) from the vertical. (a) Draw a diagram showing the forces on each sphere. Treat the spheres as point charges. (b) Find the magnitude of \(q\). (c) Both threads are now shortened to length \(L =\) 0.600 m, while the charges \(q_1\) and \(q_2\) remain unchanged. What new angle will each thread make with the vertical? (\(Hint\): This part of the problem can be solved numerically by using trial values for \(\theta\) and adjusting the values of \(\theta\) until a self-consistent answer is obtained.)

Step-by-Step Solution

Answer

For (b), the charge is approximately \( q \approx 1.4 \times 10^{-8} \text{ C} \). For (c), the new angle is approximately \( 35^\circ \).

Step 1: Analyze Forces on Spheres

When two spheres are charged and suspended, they experience several forces: gravitational force downward, tension along the thread, and electrostatic force due to charges driving them apart. Draw a Free Body Diagram (FBD) for one of the spheres. In the diagram, the tension has two components: \( T \cos(\theta) \) vertically and \( T \sin(\theta) \) horizontally. The gravitational force is \( mg \) downward, and the electrostatic force \( F_e \) is outward, perpendicular to the vertical.

Step 2: Apply Force Equilibrium Conditions

Set up the equilibrium equations using the FBD. For vertical forces: \( T \cos(\theta) = mg \). For horizontal forces: \( T \sin(\theta) = F_e \). The electrostatic force can be expressed using Coulomb's Law: \( F_e = \frac{kq^2}{r^2} \), where \( r = 2L \sin(\theta) \) is the distance between the spheres.

Step 3: Solve for Tension

Using the vertical force equilibrium equation: \( T = \frac{mg}{\cos(\theta)} \). Substituting the mass \( m = 0.015 \text{ kg} \) and \( g = 9.8 \text{ m/s}^2 \), we have: \( T = \frac{0.015 \times 9.8}{\cos(25^\circ)} \approx 0.166 \text{ N} \).

Step 4: Calculate Charge Magnitude

From the horizontal force equilibrium equation: \( T \sin(\theta) = \frac{kq^2}{r^2} \). Substitute for \( T \) and \( r \), solve for \( q \): \( q = \sqrt{\frac{T \sin(\theta) \cdot (2L \sin(\theta))^2}{k}} \). With \( k = 8.99 \times 10^9 \text{ N m}^2/\text{C}^2 \), plug in the values to find \( q \approx 1.4 \times 10^{-8} \text{ C} \).

Step 5: Setup for New Angle Calculation

After shortening the strings to \( L = 0.6 \text{ m} \), the same forces and equations apply. However, calculate trial angles \( \theta' \) until the forces balance using the updated length. The new \( r' = 2L' \sin(\theta') \). Re-calculate the horizontal force equations for different angles.

Step 6: Numerical Solution for New Configuration

Trial and error is needed to solve for the new angle \( \theta' \). Adjust \( \theta' \) and check if when inserting in the equilibrium equations with the new \( L = 0.600 \text{ m} \), both balances hold true. Through iteration, find \( \theta' \approx 35^\circ \).

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in electrostatics that describes the force between two charged objects. It states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The formula for this is: \[ F_e = \frac{kq_1q_2}{r^2} \]where \( F_e \) is the electrostatic force, \( k \) is Coulomb's constant \( (8.99 \times 10^9 \, \text{N} \, \text{m}^2/\text{C}^2) \), \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges. In the exercise, Coulomb’s Law helps determine the force pushing the charged spheres apart. Given both spheres have the same charge, the law simplifies to: \[ F_e = \frac{kq^2}{r^2} \]This relation is crucial because it allows us to calculate the magnitude of the charge \( q \) by rearranging the components of the force equilibrium equations derived from the free body diagram.

Free Body Diagram

A free body diagram (FBD) is a visual representation used in physics to illustrate the forces acting on an object. In our exercise, each sphere has several forces acting on it:- **Gravitational force**: Acts downward and is equal to \( mg \), where \( m \) is the mass of the sphere and \( g \) is the acceleration due to gravity.- **Tension force**: Acts along the silk thread. It has two components: - \( T \cos(\theta) \): Vertical component - \( T \sin(\theta) \): Horizontal component- **Electrostatic force**: Acts horizontally away from the other sphere, represented by \( F_e \).To solve the problem, it’s important to accurately draw an FBD for one of the spheres with these forces. This diagram assists in setting up the equilibrium equations, where \[ T \cos(\theta) = mg \]and\[ T \sin(\theta) = F_e \].These equations are crucial for determining the tension in the strings and the electrostatic forces required to solve for the charge on the spheres.

Force Equilibrium

In physics, force equilibrium refers to the state where all the forces acting on an object are balanced, and the object is in a stable condition. For the charged spheres, achieving force equilibrium means:

• The vertical forces (tension component and weight) must balance.
• The horizontal forces (tension component and electrostatic force) must balance.

Vertical force equilibrium can be described by the equation:\[ T \cos(\theta) = mg \]This tells us that the vertical component of the tension force must exactly counteract the force of gravity. Solving this gives us the tension:\[ T = \frac{mg}{\cos(\theta)} \].For horizontal equilibrium, the tension's horizontal component must balance the electrostatic repulsion:\[ T \sin(\theta) = \frac{kq^2}{r^2} \].By substituting \( T \) from the vertical equilibrium into the horizontal equation, we can solve for the charge \( q \). The balance of these forces ensures the spheres remain in their position. If the lengths of the strings are altered, these equations still apply but may require adjustments, such as recalculating the angle of the strings that achieve equilibrium.