---
title: "A cart of mass \\( m \\) is placed on a level, horizontal track. The cart is attached to a specialized magnetic spring setup that exerts a non-linear restoring force on the cart. The force exerted on the cart by the setup is given by the equation \\( F_s = -\\beta x^3 \\), where \\( \\beta \\) is a positive constant and \\( x \\) is the displacement of the cart from its equilibrium position at \\( x = 0 \\). Frictional forces between the cart and the track are considered to be negligible. The cart is displaced to an initial position \\( x = A \\) and released from rest."
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url: "https://nerd-notes.com/ubq/117779/"
date_modified: "2026-08-04T07:56:41+00:00"
---

# A cart of mass \( m \) is placed on a level, horizontal track. The cart is attached to a specialized magnetic spring setup that exerts a non-linear restoring force on the cart. The force exerted on the cart by the setup is given by the equation \( F_s = -\beta x^3 \), where \( \beta \) is a positive constant and \( x \) is the displacement of the cart from its equilibrium position at \( x = 0 \). Frictional forces between the cart and the track are considered to be negligible. The cart is displaced to an initial position \( x = A \) and released from rest.

A cart of mass \( m \) is placed on a level, horizontal track. The cart is attached to a specialized magnetic spring setup that exerts a non-linear restoring force on the cart. The force exerted on the cart by the setup is given by the equation \( F_s = -\beta x^3 \), where \( \beta \) is a positive constant and \( x \) is the displacement of the cart from its equilibrium position at \( x = 0 \). Frictional forces between the cart and the track are considered to be negligible. The cart is displaced to an initial position \( x = A \) and released from rest.

![A horizontal straight line representing a level track. On the track sits a rectangular cart labeled 'm'. To the left of the cart, connecting the left side of the cart to a fixed vertical wall, is a stylized coil representing the magnetic spring setup. Above the cart is a horizontal dashed axis. On this axis, a vertical tick mark aligned with the natural center of the spring setup is labeled 'x = 0'. A second vertical tick mark aligned with the center of the cart is labeled 'x = A'. An arrow points from x = 0 to x = A. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830200-jh6xj3.jpg)

**Part a)** **Indicate** whether the motion of the cart is simple harmonic motion. - [ ] Yes - [ ] No **Justify** your answer. *(2 points)*

**Part b)** Using conservation of energy, **derive** an integral expression that could be evaluated to find the period \( T \) of the cart's oscillation. Express your answer in terms of \( m \), \( \beta \), \( A \), and fundamental constants. Do not evaluate the integral. *(3 points)*

**Part c)** Students wish to experimentally determine the value of the constant \( \beta \). They have access to the cart, the magnetic spring setup, and standard laboratory equipment. **Describe** an experimental procedure the students could use to measure the period \( T \) of the cart for various initial amplitudes \( A \). Include any steps necessary to reduce experimental uncertainty. State the equipment to be used and what it will measure. *(2 points)*

**Part d)** The students perform the experiment and obtain the data shown in the table below. The integral in part (b) can be evaluated to yield the theoretical relationship \( T = \dfrac{7.42}{A} \sqrt{\dfrac{m}{\beta}} \). | Amplitude \( A \) (m) | Period \( T \) (s) |   |   | | :---: | :---: | :---: | :---: | | 0.10 | 2.10 | | | | 0.15 | 1.40 | | | | 0.20 | 1.05 | | | | 0.25 | 0.84 | | | | 0.30 | 0.70 | | |

**Part e)** **Plot** the data points for the quantities indicated in part (d)(i) on the grid provided below. Clearly scale and label all axes, including units if appropriate. **Draw** a straight line that best represents the data. *(3 points)*

**Part f)** Given that the mass of the cart is \( m = 0.20 \text{ kg} \), **calculate** an experimental value for the constant \( \beta \) using your best-fit line. *(2 points)*

**Part g)** In a second trial with a different track, the students notice that the track is not perfectly frictionless. As a result, a small amount of mechanical energy is dissipated as the cart moves, causing the maximum displacement of the cart to decrease slightly as it travels. **Predict** whether the measured time for the *first* full oscillation will be greater than, less than, or equal to the ideal period calculated assuming absolutely no friction. - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your answer using physical principles and the theoretical relationship for the period of this oscillator. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117779/*