---
title: "Two small insulating spheres, 1 and 2, are suspended from a common support point on a ceiling by lightweight insulating strings of equal length. Sphere 1 has mass \\(m_1 = m\\) and charge \\(q_1 = +Q\\), while sphere 2 has mass \\(m_2 = 2m\\) and charge \\(q_2 = +3Q\\). The spheres repel each other and reach static equilibrium with their supporting strings making angles \\(\\theta_1\\) and \\(\\theta_2\\) relative to the vertical, respectively. Which of the following correctly compares the deflection angles \\(\\theta_1\\) and \\(\\theta_2\\) and provides a valid justification?"
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url: "https://nerd-notes.com/ubq/116994/"
date_modified: "2026-08-04T06:16:58+00:00"
---

# Two small insulating spheres, 1 and 2, are suspended from a common support point on a ceiling by lightweight insulating strings of equal length. Sphere 1 has mass \(m_1 = m\) and charge \(q_1 = +Q\), while sphere 2 has mass \(m_2 = 2m\) and charge \(q_2 = +3Q\). The spheres repel each other and reach static equilibrium with their supporting strings making angles \(\theta_1\) and \(\theta_2\) relative to the vertical, respectively. Which of the following correctly compares the deflection angles \(\theta_1\) and \(\theta_2\) and provides a valid justification?

Two small insulating spheres, 1 and 2, are suspended from a common support point on a ceiling by lightweight insulating strings of equal length. Sphere 1 has mass \(m_1 = m\) and charge \(q_1 = +Q\), while sphere 2 has mass \(m_2 = 2m\) and charge \(q_2 = +3Q\). The spheres repel each other and reach static equilibrium with their supporting strings making angles \(\theta_1\) and \(\theta_2\) relative to the vertical, respectively. Which of the following correctly compares the deflection angles \(\theta_1\) and \(\theta_2\) and provides a valid justification?

![Two light insulating strings of identical length hang from a single support point on a horizontal ceiling. The left string holds a small sphere labeled 1 with mass m and charge +Q, inclined at an angle theta1 relative to a central vertical dashed reference line. The right string holds a small sphere labeled 2 with mass 2m and charge +3Q, inclined at an angle theta2 relative to the same central vertical dashed reference line. Both spheres are rendered symmetrically with respect to the vertical reference line. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785824217-tXEbyO.jpg)

- **A.** \(\theta_1 < \theta_2\), because sphere 2 has a larger charge of \(+3Q\) and thus exerts a greater electrostatic force on sphere 1 than sphere 1 exerts on sphere 2.
- **B.** \(\theta_1 < \theta_2\), because sphere 2 has greater mass, requiring a larger deflection angle to establish equilibrium against gravity.
- **C.** \(\theta_1 = \theta_2\), because by Newton's third law the electrostatic forces on the two spheres are equal in magnitude, so their deflection angles must also be equal.
- **D.** \(\theta_1 > \theta_2\), because the electrostatic forces on both spheres are equal in magnitude, but sphere 1 has a smaller mass, resulting in a larger deflection angle to satisfy equilibrium.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/116994/*