---
title: "A small sphere of mass \\(m\\) is attached to one end of a light, inextensible string of length \\(L\\). The other end of the string is attached to a point on a rigid vertical pole. The sphere is set into motion such that it travels in a horizontal circle at a constant speed \\(v\\). As the sphere swings, the string maintains a constant angle \\(\\theta\\) with the vertical pole, as shown in Figure 1. Frictional forces and air resistance are negligible."
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url: "https://nerd-notes.com/ubq/110354/"
date_modified: "2026-04-02T07:10:54+00:00"
---

# A small sphere of mass \(m\) is attached to one end of a light, inextensible string of length \(L\). The other end of the string is attached to a point on a rigid vertical pole. The sphere is set into motion such that it travels in a horizontal circle at a constant speed \(v\). As the sphere swings, the string maintains a constant angle \(\theta\) with the vertical pole, as shown in Figure 1. Frictional forces and air resistance are negligible.

A small sphere of mass \(m\) is attached to one end of a light, inextensible string of length \(L\). The other end of the string is attached to a point on a rigid vertical pole. The sphere is set into motion such that it travels in a horizontal circle at a constant speed \(v\). As the sphere swings, the string maintains a constant angle \(\theta\) with the vertical pole, as shown in Figure 1. Frictional forces and air resistance are negligible.

![A vertical pole is anchored to a flat surface. A dashed horizontal ellipse indicates the circular path of the sphere in perspective. A solid line representing a string connects a point on the vertical pole down to a solid dot representing the sphere. An angle arc and label '\(\theta\)' indicate the angle between the vertical pole and the string. The string is labeled with length '\(L\)'. The sphere is labeled with mass '\(m\)'. An arrow tangent to the dashed circle points forward to indicate the velocity '\(v\)'.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1775095745-Obd7Dx.jpg)

**Part a)** **Draw and label** a free-body diagram representing the forces exerted on the sphere when it is at the position shown in Figure 1. Represent each force by a distinct arrow starting on, and pointing away from, the dot. *(2 points)*

**Part b)** **Derive** an expression for the tension \(F_T\) in the string. Express your answer in terms of \(m\), \(\theta\), and fundamental constants. *(2 points)*

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

**Part d)** The original string is replaced by a new string of length \(4L\). The original sphere of mass \(m\) is spun so that the angle \(\theta\) remains the exact same as in the original experiment. **Determine** the ratio of the new speed \(v_{new}\) to the original speed \(v\). **Justify** your answer using your derived expression from part (c). *(2 points)*

**Part e)** The original string of length \(L\) is used again, but the sphere is replaced by a heavier sphere of mass \(2m\). The heavier sphere is spun so that it travels at the exact same constant speed \(v\) as the original sphere. **Indicate** whether the new angle that the string makes with the vertical is greater than, less than, or equal to the original angle \(\theta\). - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your answer using physical principles and functional dependence. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/110354/*