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title: "An experiment is designed to determine the angular frequency \\(\\omega\\) of a horizontal oscillator consisting of a block of mass \\(M\\) attached to a spring of force constant \\(k\\), length \\(L\\), and uniform mass \\(m_s\\). The block is released from rest from various displacement amplitudes \\(A\\) along a frictionless horizontal track, and its maximum speed \\(v_{\\text{max}}\\) is recorded for each trial. Each element of the spring at a distance \\(x\\) from its fixed end moves with speed \\(v(x) = \\left(\\dfrac{x}{L}\\right)v\\), where \\(v\\) is the instantaneous speed of the block. If the student plots \\(v_{\\text{max}}\\) on the vertical axis as a function of \\(A\\) on the horizontal axis, which of the following correctly identifies the theoretical slope of the best-fit line and the effect of the spring’s mass on the graph compared to that of an ideal massless spring?"
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url: "https://nerd-notes.com/ubq/124671/"
date_modified: "2026-09-28T14:09:02+00:00"
---

# An experiment is designed to determine the angular frequency \(\omega\) of a horizontal oscillator consisting of a block of mass \(M\) attached to a spring of force constant \(k\), length \(L\), and uniform mass \(m_s\). The block is released from rest from various displacement amplitudes \(A\) along a frictionless horizontal track, and its maximum speed \(v_{\text{max}}\) is recorded for each trial. Each element of the spring at a distance \(x\) from its fixed end moves with speed \(v(x) = \left(\dfrac{x}{L}\right)v\), where \(v\) is the instantaneous speed of the block. If the student plots \(v_{\text{max}}\) on the vertical axis as a function of \(A\) on the horizontal axis, which of the following correctly identifies the theoretical slope of the best-fit line and the effect of the spring’s mass on the graph compared to that of an ideal massless spring?

An experiment is designed to determine the angular frequency \(\omega\) of a horizontal oscillator consisting of a block of mass \(M\) attached to a spring of force constant \(k\), length \(L\), and uniform mass \(m_s\). The block is released from rest from various displacement amplitudes \(A\) along a frictionless horizontal track, and its maximum speed \(v_{\text{max}}\) is recorded for each trial. Each element of the spring at a distance \(x\) from its fixed end moves with speed \(v(x) = \left(\dfrac{x}{L}\right)v\), where \(v\) is the instantaneous speed of the block. If the student plots \(v_{\text{max}}\) on the vertical axis as a function of \(A\) on the horizontal axis, which of the following correctly identifies the theoretical slope of the best-fit line and the effect of the spring's mass on the graph compared to that of an ideal massless spring?

![A horizontal spring-mass system resting on a horizontal surface. On the left is a vertical wall shown with diagonal hash marks along its left boundary. A horizontal helical spring extends from the wall to a square block of mass \(M\) on the right. The block rests on a horizontal line representing a frictionless floor. Above the spring, a horizontal double-headed dimension arrow spans the full length of the spring from the vertical wall to the left face of the block, labeled \(L\). A horizontal axis below the floor has an origin tick labeled 0 at the wall and points to the right with an arrow labeled \(x\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604542-OegdaN.jpg)

- **A.** The slope is \(\sqrt{\dfrac{k}{M + m_s}}\), which is smaller than for an ideal spring, and the vertical intercept remains zero.
- **B.** The slope is \(\sqrt{\dfrac{k}{M + \dfrac{1}{3}m_s}}\), which is smaller than for an ideal spring, and the vertical intercept remains zero.
- **C.** The slope is \(\sqrt{\dfrac{k}{M + \dfrac{1}{3}m_s}}\), which is smaller than for an ideal spring, and the vertical intercept is shifted to a positive value.
- **D.** The slope is \(\sqrt{\dfrac{k}{M + \dfrac{1}{2}m_s}}\), which is smaller than for an ideal spring, and the vertical intercept remains zero.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124671/*