---
title: "A cart of known mass \\(m\\) is placed on a horizontal track. One end of a thick rubber band is attached to a fixed vertical wall at the left end of the track, and the other end is attached to the cart, as shown in Figure 1. A student claims that the rubber band does not obey Hooke’s law, but instead stores elastic potential energy according to the equation \\(U = C x^3\\), where \\(x\\) is the stretch distance of the rubber band from its unstretched equilibrium position and \\(C\\) is a constant. The student wants to conduct an experiment to test this claim and determine the value of the constant \\(C\\). Assume the track has negligible friction and the mass of the rubber band is negligible."
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url: "https://nerd-notes.com/ubq/109223/"
date_modified: "2026-03-22T10:13:26+00:00"
---

# A cart of known mass \(m\) is placed on a horizontal track. One end of a thick rubber band is attached to a fixed vertical wall at the left end of the track, and the other end is attached to the cart, as shown in Figure 1. A student claims that the rubber band does not obey Hooke’s law, but instead stores elastic potential energy according to the equation \(U = C x^3\), where \(x\) is the stretch distance of the rubber band from its unstretched equilibrium position and \(C\) is a constant. The student wants to conduct an experiment to test this claim and determine the value of the constant \(C\). Assume the track has negligible friction and the mass of the rubber band is negligible.

A cart of known mass \(m\) is placed on a horizontal track. One end of a thick rubber band is attached to a fixed vertical wall at the left end of the track, and the other end is attached to the cart, as shown in Figure 1. A student claims that the rubber band does not obey Hooke's law, but instead stores elastic potential energy according to the equation \(U = C x^3\), where \(x\) is the stretch distance of the rubber band from its unstretched equilibrium position and \(C\) is a constant. The student wants to conduct an experiment to test this claim and determine the value of the constant \(C\). Assume the track has negligible friction and the mass of the rubber band is negligible.

![A horizontal straight line representing a track. On the track is a rectangular box representing a cart. To the left of the cart is a vertical line representing a wall. Connecting the wall to the left side of the cart is a thick wavy line representing a rubber band. Above the rubber band is a horizontal double-headed arrow labeled with the variable \(x\), indicating the distance the rubber band is stretched.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774091520-0C8TNz.jpg)

**Part a)** **Design** an experimental procedure to test the student's claim that the potential energy stored in the rubber band is given by \(U = C x^3\). *(4 points)*

**Part b)** The student collects the data and wants to create a linear graph to verify the relationship \(U = C x^3\) and determine the constant \(C\). *(3 points)*

**Part c)** Another student argues that the rubber band actually behaves like an ideal spring, storing potential energy according to \(U = \dfrac{1}{2}kx^2\). Suppose this second student is correct. If the first student still plots the quantities identified in part (b)(i), **predict** whether the resulting graph will be a straight line, concave up, or concave down. **Justify** your answer using physical principles. *(3 points)*

**Part d)** The rubber band is now removed. The cart is placed on a different horizontal track that has non-negligible friction. The cart is pushed against an unattached ideal spring of known spring constant \(k\), compressing the spring a distance \(D\). The cart is then released from rest. The cart loses contact with the spring and travels a total distance \(L\) from its release point before coming to a complete stop. **Derive** an expression for the coefficient of kinetic friction \(\mu_k\) between the cart and the track in terms of \(m\), \(k\), \(D\), \(L\), and fundamental constants. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109223/*