---
title: "A block of mass \\(m\\) is attached to an ideal horizontal spring with spring constant \\(k\\). The block rests on a frictionless horizontal surface. The block is pulled to a positive position \\(x = A\\) and released from rest so that it oscillates.   Figure 1 shows the kinetic energy \\(K\\) of the block as a function of its position \\(x\\) for one full cycle. The maximum kinetic energy of the block is labeled \\(K_0\\)."
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url: "https://nerd-notes.com/ubq/109393/"
date_modified: "2026-03-21T23:46:14+00:00"
---

# A block of mass \(m\) is attached to an ideal horizontal spring with spring constant \(k\). The block rests on a frictionless horizontal surface. The block is pulled to a positive position \(x = A\) and released from rest so that it oscillates. 

Figure 1 shows the kinetic energy \(K\) of the block as a function of its position \(x\) for one full cycle. The maximum kinetic energy of the block is labeled \(K_0\).

A block of mass \(m\) is attached to an ideal horizontal spring with spring constant \(k\). The block rests on a frictionless horizontal surface. The block is pulled to a positive position \(x = A\) and released from rest so that it oscillates. 

Figure 1 shows the kinetic energy \(K\) of the block as a function of its position \(x\) for one full cycle. The maximum kinetic energy of the block is labeled \(K_0\).

![A two-dimensional Cartesian graph. The vertical axis is labeled 'Kinetic Energy, K' with an arrow pointing upward. The horizontal axis is labeled 'Position, x' with an arrow pointing right. The origin is marked '0'. A solid line curve in the shape of a downward-opening parabola is drawn. The vertex (peak) of the parabola lies on the positive y-axis and is labeled with a tick mark 'K_0'. The parabola crosses the x-axis at two locations. The left x-intercept has a tick mark labeled '-A' and the right x-intercept has a tick mark labeled 'A'.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774136773-HuA1dA.jpg)

**Part a)** **Derive** an expression for the spring constant \(k\) in terms of \(K_0\), \(A\), and fundamental constants as appropriate. *(2 points)*

**Part b)** A student claims that if the experiment is repeated using a lighter block of mass \(m/2\), with the same spring constant \(k\) and same initial displacement \(A\), the maximum speed of the block will double because the mass is halved. *(5 points)*

**Part c)** The original block of mass \(m\) and spring \(k\) are used again, but the block is now displaced to a greater initial position \(x = 2A\). On the blank axes provided, **sketch** the graph of the kinetic energy \(K\) as a function of position \(x\) for this new experiment. *(3 points)*

**Part d)** Another student proposes the following equation for the velocity \(v\) of the block as a function of time \(t\) for the experiment described in Part (c) (where the initial displacement is \(2A\)): \[ v(t) = -2A \left( \dfrac{k}{m} \right) \sin\left( \sqrt{\dfrac{k}{m}} t \right) \] **Indicate** whether this proposed equation is physically plausible. - [ ] Plausible - [ ] Not plausible **Justify** your reasoning by evaluating the physical dimensions (units) or functional dependence of the proposed equation. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109393/*