---
title: "A cart of mass \\( m_0 \\) oscillates on a horizontal, frictionless track while attached to an ideal spring of unknown spring constant \\( k \\). The equilibrium position of the system is at \\( x = 0 \\). Figure 1 shows a graph of the cart’s kinetic energy \\( K \\) as a function of its position \\( x \\). The maximum kinetic energy is \\( K_0 \\), and the cart oscillates between a minimum position \\( x = -D \\) and a maximum position \\( x = +D \\)."
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url: "https://nerd-notes.com/ubq/109583/"
date_modified: "2026-03-25T07:05:56+00:00"
---

# A cart of mass \( m_0 \) oscillates on a horizontal, frictionless track while attached to an ideal spring of unknown spring constant \( k \). The equilibrium position of the system is at \( x = 0 \). Figure 1 shows a graph of the cart’s kinetic energy \( K \) as a function of its position \( x \). The maximum kinetic energy is \( K_0 \), and the cart oscillates between a minimum position \( x = -D \) and a maximum position \( x = +D \).

A cart of mass \( m_0 \) oscillates on a horizontal, frictionless track while attached to an ideal spring of unknown spring constant \( k \). The equilibrium position of the system is at \( x = 0 \). Figure 1 shows a graph of the cart's kinetic energy \( K \) as a function of its position \( x \). The maximum kinetic energy is \( K_0 \), and the cart oscillates between a minimum position \( x = -D \) and a maximum position \( x = +D \).

![A line graph plotted on a Cartesian coordinate system. The horizontal axis is labeled 'Position \( x \)' and the vertical axis is labeled 'Kinetic Energy \( K \)'. The origin (0,0) is marked. A smooth, downward-opening (concave down) parabolic curve is drawn. The peak (vertex) of the parabola lies on the vertical axis at a marked value of \( K_0 \). The parabola intersects the horizontal axis at two symmetric points marked \( -D \) and \( +D \).](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774422356-87eDQP.jpg)

**Part a)** A student looking at the graph states, "The graph shows that the kinetic energy decreases as the cart moves from \( x = 0 \) to \( x = D \). Because the kinetic energy is decreasing, the acceleration of the cart must be decreasing in magnitude as it approaches \( x = D \)." **Identify** the error in the student's reasoning, and **justify** the correct relationship between the cart's position and the magnitude of its acceleration using physical principles. *(3 points)*

**Part b)** **Derive** an expression for the spring constant \( k \) of the spring. Express your answer in terms of \( m_0 \), \( K_0 \), \( D \), and fundamental constants as appropriate. *(2 points)*

**Part c)** **Derive** an expression for the period \( T \) of the cart's oscillation. Express your answer in terms of \( m_0 \), \( K_0 \), \( D \), and fundamental constants as appropriate. *(2 points)*

**Part d)** The original spring is replaced by a new spring that has a larger spring constant \( k_{new} \) (where \( k_{new} > k \)). The same cart of mass \( m_0 \) is displaced to the same initial position \( x = D \) and released from rest. Two students make claims about the new maximum speed of the cart: *   **Student Y:** "Because the amplitude \( D \) is the same, the cart will have the same maximum kinetic energy, and therefore its maximum speed will be the same as before." *   **Student Z:** "The new spring is stiffer, so it will exert a larger restoring force on the cart for any given displacement. This will cause the cart to reach a greater maximum speed at the equilibrium position." *(5 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109583/*