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title: "An ideal horizontal spring with spring constant \\(k\\) is attached to a block of mass \\(m\\) on a frictionless surface. At time \\(t = 0\\), the block is at displacement \\(x_0 > 0\\) and moving with velocity \\(v_0 > 0\\), resulting in simple harmonic motion described by \\(x(t) = A\\cos(\\omega t + \\phi)\\), where \\(\\omega = \\sqrt{k/m}\\). An experimenter observes that scaling both \\(x_0\\) and \\(v_0\\) by a positive factor \\(\\beta\\) scales the resulting amplitude \\(A\\) by \\(\\beta\\) and leaves the phase constant \\(\\phi\\) unchanged, whereas scaling only \\(x_0\\) by \\(\\beta\\) changes both \\(A\\) and \\(\\phi\\). Which of the following statements provides the correct physical explanation for these observations?"
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url: "https://nerd-notes.com/ubq/121007/"
date_modified: "2026-08-23T04:45:09+00:00"
---

# An ideal horizontal spring with spring constant \(k\) is attached to a block of mass \(m\) on a frictionless surface. At time \(t = 0\), the block is at displacement \(x_0 > 0\) and moving with velocity \(v_0 > 0\), resulting in simple harmonic motion described by \(x(t) = A\cos(\omega t + \phi)\), where \(\omega = \sqrt{k/m}\). An experimenter observes that scaling both \(x_0\) and \(v_0\) by a positive factor \(\beta\) scales the resulting amplitude \(A\) by \(\beta\) and leaves the phase constant \(\phi\) unchanged, whereas scaling only \(x_0\) by \(\beta\) changes both \(A\) and \(\phi\). Which of the following statements provides the correct physical explanation for these observations?

An ideal horizontal spring with spring constant \(k\) is attached to a block of mass \(m\) on a frictionless surface. At time \(t = 0\), the block is at displacement \(x_0 > 0\) and moving with velocity \(v_0 > 0\), resulting in simple harmonic motion described by \(x(t) = A\cos(\omega t + \phi)\), where \(\omega = \sqrt{k/m}\). An experimenter observes that scaling both \(x_0\) and \(v_0\) by a positive factor \(\beta\) scales the resulting amplitude \(A\) by \(\beta\) and leaves the phase constant \(\phi\) unchanged, whereas scaling only \(x_0\) by \(\beta\) changes both \(A\) and \(\phi\). Which of the following statements provides the correct physical explanation for these observations?

![A horizontal frictionless surface with a vertical wall on the left. A horizontal coiled spring extends from the wall to a rectangular block of mass \(m\). A vertical dashed reference line labeled \(x = 0\) indicates the equilibrium position. The block is positioned to the right of the dashed line, with a horizontal dimension arrow from \(x = 0\) to the center of the block labeled \(x_0\). A horizontal vector arrow pointing to the right originates from the right face of the block and is labeled \(v_0\). The spring has exactly six coils and is labeled with spring constant \(k\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460309-JM4Fot.jpg)

- **A.** The phase constant is governed by the time offset \(\Delta t = x_0 / v_0\) required for the block to reach the equilibrium position under its initial velocity, so scaling both \(x_0\) and \(v_0\) by \(\beta\) preserves this transit time ratio, whereas scaling \(x_0\) alone increases the transit time and shifts \(\phi\).
- **B.** The amplitude is given by the linear superposition \(A = x_0 + v_0/\omega\), and scaling \(x_0\) alone alters the effective spring restoring force and changes the oscillator's natural frequency \(\omega\), whereas scaling both parameters by \(\beta\) preserves the natural frequency and leaves \(\phi\) unchanged.
- **C.** The total mechanical energy is \(E = \dfrac{1}{2}kx_0^2 + \dfrac{1}{2}mv_0^2\), and the phase constant is the fixed angle \(\phi = \arctan(kx_0 / mv_0)\) representing the ratio of maximum spring force to initial momentum; scaling \(x_0\) alone alters this force-to-momentum ratio, while scaling both changes the total energy without altering the dynamic ratio.
- **D.** The general solution can be written as \(x(t) = x_0\cos(\omega t) + \dfrac{v_0}{\omega}\sin(\omega t)\), which yields an amplitude \(A = \sqrt{x_0^2 + (v_0/\omega)^2}\) and a phase constant satisfying \(\tan\phi = -\dfrac{v_0}{\omega x_0}\); scaling both \(x_0\) and \(v_0\) by \(\beta\) scales the amplitude by \(\sqrt{(\beta x_0)^2 + (\beta v_0/\omega)^2} = \beta A\) and preserves the ratio \(\dfrac{v_0}{x_0}\), whereas scaling \(x_0\) alone changes both the square-root sum and the ratio.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121007/*