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Mass-Spring Oscillation Period

A public oscillation question showing how mass and spring constant set the period of simple harmonic motion independent of amplitude.

Question

A 0.50 kg0.50\,\text{kg} block attached to an ideal spring with spring constant k=200 N/mk = 200\,\text{N/m} oscillates on a frictionless horizontal surface. What is the period of the oscillation?

  1. a.0.05 s0.05\,\text{s}
  2. b.0.16 s0.16\,\text{s}
  3. c.0.31 s0.31\,\text{s}
  4. d.3.2 s3.2\,\text{s}

Correct answer

C. 0.31 s0.31\,\text{s}

Full reasoning

  1. 1Compute the ratio under the root: m/k=0.50/200=0.0025 s2m/k = 0.50/200 = 0.0025\,\text{s}^2.
  2. 2Take the square root: 0.0025=0.05 s\sqrt{0.0025} = 0.05\,\text{s}, which is the reciprocal of the angular frequency ω=20 rad/s\omega = 20\,\text{rad/s}.
  3. 3Multiply by 2π2\pi for a full cycle: T=2π(0.05)≈0.31 sT = 2\pi(0.05) \approx 0.31\,\text{s}.
  4. 4Amplitude appears nowhere in the formula, so the period is amplitude-independent.

Why each choice is right or wrong

Choice A

This is m/k\sqrt{m/k} alone; the factor of 2π2\pi that converts the angular timescale to a full cycle was dropped.

Choice B

This is the half period — the time from one turning point to the opposite turning point — not a full cycle.

Choice C

Correct. T=2π0.50/200=2π(0.05)≈0.31 sT = 2\pi\sqrt{0.50/200} = 2\pi(0.05) \approx 0.31\,\text{s} for one complete oscillation.

Choice D

This is the frequency, about 3.2 Hz3.2\,\text{Hz}, mislabeled as a period instead of taking its reciprocal.

Related formula

Period of a mass-spring oscillator

T=2πm/kT = 2\pi\sqrt{m/k}

Assumptions: The spring is ideal (massless and linear, obeying Hooke's law). No damping acts on the oscillator.

Related topic and practice

Simple Harmonic MotionClassical Mechanics Domain GuideEnergy and Momentum ConservationNewton's Second Law and Free-Body DiagramsA collision that conserves momentum but loses kinetic energyEnergy Conservation on a Frictionless TrackCreate a free account to practice

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Sources

Authored for this public explainer registry. It has no database question ID and is not copied from a protected or official exam bank.

  1. University Physics Volume 1, Section 15.1: Simple Harmonic Motion — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
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Sources last checked 2026-08-07

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