---
title: "A block of mass \\(m\\) rests on a horizontal frictionless surface and is attached to a rigid vertical wall by two ideal, massless springs connected in series. The springs have spring constants \\(k_1\\) and \\(k_2\\), where \\(k_1\\) is held fixed. The block is pulled horizontally, displacing both springs from their equilibrium lengths, and released from rest so that it undergoes simple harmonic motion with period \\(T\\). Which of the following correctly identifies the period of oscillation \\(T\\) in the limit as the second spring becomes infinitely stiff (\\(k_2 \\to \\infty\\)), and provides the correct physical justification?"
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url: "https://nerd-notes.com/ubq/124567/"
date_modified: "2026-09-28T14:08:41+00:00"
---

# A block of mass \(m\) rests on a horizontal frictionless surface and is attached to a rigid vertical wall by two ideal, massless springs connected in series. The springs have spring constants \(k_1\) and \(k_2\), where \(k_1\) is held fixed. The block is pulled horizontally, displacing both springs from their equilibrium lengths, and released from rest so that it undergoes simple harmonic motion with period \(T\). Which of the following correctly identifies the period of oscillation \(T\) in the limit as the second spring becomes infinitely stiff (\(k_2 \to \infty\)), and provides the correct physical justification?

A block of mass \(m\) rests on a horizontal frictionless surface and is attached to a rigid vertical wall by two ideal, massless springs connected in series. The springs have spring constants \(k_1\) and \(k_2\), where \(k_1\) is held fixed. The block is pulled horizontally, displacing both springs from their equilibrium lengths, and released from rest so that it undergoes simple harmonic motion with period \(T\). Which of the following correctly identifies the period of oscillation \(T\) in the limit as the second spring becomes infinitely stiff (\(k_2 \to \infty\)), and provides the correct physical justification?

![A vertical cross-hatched support wall is shown on the left side, joined at a right angle to a flat horizontal surface that extends to the right. Attached to the vertical wall is a horizontal coiled spring labeled \(k_1\). The right end of this first spring connects at a single junction point to the left end of a second horizontal coiled spring labeled \(k_2\), aligned along the same horizontal line. The right end of the second spring attaches to the center of the left vertical face of a solid rectangular block labeled \(m\) that rests on the horizontal surface. No other labels, lines, text, or arrows appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604521-uPEsqy.jpg)

- **A.** \(T \to 0\), because the effective spring constant is the sum of the two constants (\(k_{\text{eff}} = k_1 + k_2\)), so the restoring force becomes infinitely large for any displacement.
- **B.** \(T \to 0\), because the stiffer spring dominates the combination, requiring an arbitrarily large force to displace the block and driving the oscillation frequency to infinity.
- **C.** \(T \to 2\pi\sqrt{\dfrac{m}{k_1}}\), because the stiffer spring stores virtually all of the system's mechanical energy, causing the softer spring to undergo negligible elongation.
- **D.** \(T \to 2\pi\sqrt{\dfrac{m}{k_1}}\), because both springs experience the same tension, causing the elongation of the stiffer spring to approach zero so that it behaves as a rigid connection.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124567/*