---
title: "A solid cylinder of mass \\(M\\) and radius \\(R\\) has a rotational inertia \\(I = \\dfrac{1}{2}MR^2\\). The cylinder is mounted on a fixed, frictionless horizontal axle that passes through its center. A nonlinear torsion spring is connected to the axle. When the cylinder is rotated by an angle \\(\\theta\\) from its equilibrium position at \\(\\theta = 0\\), the spring exerts a restoring torque \\(\\tau\\) on the cylinder given by the function \\(\\tau(\\theta) = -C \\theta^3\\), where \\(C\\) is a positive constant. The cylinder is rotated to a positive initial angle \\(\\theta_0\\) and held at rest."
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url: "https://nerd-notes.com/ubq/117801/"
date_modified: "2026-08-04T07:57:07+00:00"
---

# A solid cylinder of mass \(M\) and radius \(R\) has a rotational inertia \(I = \dfrac{1}{2}MR^2\). The cylinder is mounted on a fixed, frictionless horizontal axle that passes through its center. A nonlinear torsion spring is connected to the axle. When the cylinder is rotated by an angle \(\theta\) from its equilibrium position at \(\theta = 0\), the spring exerts a restoring torque \(\tau\) on the cylinder given by the function \(\tau(\theta) = -C \theta^3\), where \(C\) is a positive constant. The cylinder is rotated to a positive initial angle \(\theta_0\) and held at rest.

A solid cylinder of mass \(M\) and radius \(R\) has a rotational inertia \(I = \dfrac{1}{2}MR^2\). The cylinder is mounted on a fixed, frictionless horizontal axle that passes through its center. A nonlinear torsion spring is connected to the axle. When the cylinder is rotated by an angle \(\theta\) from its equilibrium position at \(\theta = 0\), the spring exerts a restoring torque \(\tau\) on the cylinder given by the function \(\tau(\theta) = -C \theta^3\), where \(C\) is a positive constant. The cylinder is rotated to a positive initial angle \(\theta_0\) and held at rest.

![A solid uniform cylinder viewed in 3D isometric perspective, mounted on a horizontal axle passing through its circular center. The cylinder has a label \(M\), \(R\). A spiral torsion spring is wrapped around the axle; one end of the spring connects to the axle and the other end connects to a fixed, rigid vertical support wall nearby. A curved arrow starting from a horizontal dashed reference line points along the circular face of the cylinder to indicate an angular displacement, labeled \(\theta\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830226-l3TMaJ.jpg)

**Part a)** Represent the characteristics of the nonlinear spring graphically. *(3 points)*

**Part b)** The cylinder is released from rest at \(\theta = \theta_0\). **Derive** an expression for the work done by the torsion spring on the cylinder as it rotates from \(\theta = \theta_0\) to \(\theta = 0\). Express your answer in terms of \(C\), \(\theta_0\), and fundamental constants. *(2 points)*

**Part c)** **Derive** an expression for the maximum angular speed of the cylinder after it is released. Express your answer in terms of \(M\), \(R\), \(C\), \(\theta_0\), and fundamental constants. *(3 points)*

**Part d)** The nonlinear spring is replaced by a standard linear torsion spring that exerts a restoring torque \(\tau_L = -\kappa\theta\). The constant \(\kappa\) is chosen such that the linear spring stores the exact same total potential energy at \(\theta_0\) as the nonlinear spring. The cylinder is again released from rest at \(\theta_0\). **Compare** the rotational kinetic energy of the cylinder with the nonlinear spring, \(K_{NL}\), to the rotational kinetic energy of the cylinder with the linear spring, \(K_L\), when the cylinder passes through the angle \(\theta = \dfrac{1}{2}\theta_0\). - [ ] \(K_{NL} > K_L\) - [ ] \(K_{NL} < K_L\) - [ ] \(K_{NL} = K_L\) **Justify** your answer using features of your graphs from part (a) or other physical principles. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117801/*