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title: "An ideal vertical mass-spring system consists of a block of mass \\(m\\) suspended from an ideal spring of force constant \\(k\\). When mounted inside an elevator at rest, the block oscillates vertically with period \\(T_0\\) about its equilibrium position. The elevator is then given a constant upward acceleration \\(a_0\\), and the block continues to execute simple harmonic motion about a new equilibrium position. An observer finds that the period of oscillation in the accelerating elevator remains equal to \\(T_0\\). Which of the following statements provides the correct physical explanation for why the period is independent of the upward acceleration?"
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url: "https://nerd-notes.com/ubq/120962/"
date_modified: "2026-08-23T04:44:46+00:00"
---

# An ideal vertical mass-spring system consists of a block of mass \(m\) suspended from an ideal spring of force constant \(k\). When mounted inside an elevator at rest, the block oscillates vertically with period \(T_0\) about its equilibrium position. The elevator is then given a constant upward acceleration \(a_0\), and the block continues to execute simple harmonic motion about a new equilibrium position. An observer finds that the period of oscillation in the accelerating elevator remains equal to \(T_0\). Which of the following statements provides the correct physical explanation for why the period is independent of the upward acceleration?

An ideal vertical mass-spring system consists of a block of mass \(m\) suspended from an ideal spring of force constant \(k\). When mounted inside an elevator at rest, the block oscillates vertically with period \(T_0\) about its equilibrium position. The elevator is then given a constant upward acceleration \(a_0\), and the block continues to execute simple harmonic motion about a new equilibrium position. An observer finds that the period of oscillation in the accelerating elevator remains equal to \(T_0\). Which of the following statements provides the correct physical explanation for why the period is independent of the upward acceleration?

- **A.** The upward acceleration increases the effective gravitational field to \(g + a_0\), but this increases both the magnitude of the restoring force and the effective inertial resistance of the oscillating mass by the same factor, leaving the ratio \(\dfrac{F_{\text{net}}}{m}\) invariant throughout the motion.
- **B.** The constant inertial force and gravity combine to shift the equilibrium position of the system, but the net restoring force on the block remains strictly proportional to its displacement from this new equilibrium with proportionality constant \(k\), keeping the governing differential equation \(\dfrac{d^2u}{dt^2} + \dfrac{k}{m}u = 0\) unchanged.
- **C.** The upward acceleration of the elevator performs net positive work on the oscillating system each cycle, increasing the total mechanical energy such that the greater average speed of the block exactly offsets the increased path length traveled relative to the ground.
- **D.** The effective downward acceleration \(g + a_0\) speeds up the block during the downward portion of each cycle, while the increased maximum spring tension speeds up the block by an identical amount during the upward portion, yielding symmetric time changes that cancel over a full period.

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