---
title: "A sphere of mass \\(m\\) is attached to one end of a string of length \\(L\\). The other end of the string is attached to a fixed pivot point on a ceiling. The sphere is set into motion such that it moves with a constant speed \\(v\\) in a horizontal circle, forming a conical pendulum. The string maintains a constant angle \\(\\theta\\) with the vertical, as shown in Figure 1. The acceleration due to gravity is \\(g\\). Air resistance and the mass of the string are negligible."
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url: "https://nerd-notes.com/ubq/109296/"
date_modified: "2026-03-21T23:11:17+00:00"
---

# A sphere of mass \(m\) is attached to one end of a string of length \(L\). The other end of the string is attached to a fixed pivot point on a ceiling. The sphere is set into motion such that it moves with a constant speed \(v\) in a horizontal circle, forming a conical pendulum. The string maintains a constant angle \(\theta\) with the vertical, as shown in Figure 1. The acceleration due to gravity is \(g\). Air resistance and the mass of the string are negligible.

A sphere of mass \(m\) is attached to one end of a string of length \(L\). The other end of the string is attached to a fixed pivot point on a ceiling. The sphere is set into motion such that it moves with a constant speed \(v\) in a horizontal circle, forming a conical pendulum. The string maintains a constant angle \(\theta\) with the vertical, as shown in Figure 1. The acceleration due to gravity is \(g\). Air resistance and the mass of the string are negligible.

![A 2D illustration of a conical pendulum. A horizontal line at the top represents a ceiling. A vertical dashed line extends straight down from a pivot point on the ceiling. A solid line representing a string extends downwards and to the right from the pivot point at an angle relative to the vertical dashed line. A small solid sphere labeled 'm' is attached to the bottom end of the string. A dashed ellipse is drawn in perspective horizontally around the vertical dashed line, passing through the sphere, to represent the horizontal circular path of the sphere. The string is labeled 'L'. An arc between the vertical dashed line and the string is labeled '\theta'.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774134676-6av2zh.jpg)

**Part a)** The dot below represents the sphere. **Draw** and **label** the forces (not components) that are exerted on the sphere. Draw the relative lengths of all vectors to reflect the relative magnitudes of all the forces. *(2 points)*

**Part b)** **Derive** an expression for the speed \(v\) of the sphere. Express your answer in terms of \(m\), \(L\), \(\theta\), and fundamental constants, as appropriate. *(3 points)*

**Part c)** **Derive** an expression for the time it takes the sphere to complete one full revolution (the period \(P\)). Express your answer in terms of \(m\), \(L\), \(\theta\), and fundamental constants, as appropriate. *(2 points)*

**Part d)** As the sphere is made to swing at larger and larger constant angles \(\theta\) (where \(0^\circ \le \theta < 90^\circ\)), the tension in the string changes. **Sketch** a graph of the magnitude of the tension force \(F_T\) as a function of the angle \(\theta\). Explicitly label the \(F_T\)-intercept on the vertical axis in terms of given variables and fundamental constants. *(2 points)*

**Part e)** The entire setup is transported to a new planet where the acceleration due to gravity is greater than \(g\). The sphere is set into a conical pendulum motion such that it maintains the **same** angle \(\theta\) as it did on Earth. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109296/*