Derive spring period and distinguish frequency measures

1. Frame the question

One cycle returns the oscillator to the same position and direction of motion. The period T is time per cycle; frequency f=1/T counts cycles per second. Angular frequency ω=2πf measures phase advance in radians per second. Confusing f and ω introduces a factor of 2π, even though both describe the same repetition.

2. Build the mechanism

For the ideal spring, a=−(k/m)x. A sinusoidal motion has a=−ω²x, so matching coefficients gives ω=√(k/m), T=2π√(m/k) and f=(1/2π)√(k/m). The square root is essential: quadrupling mass doubles the period, while quadrupling stiffness halves it. Units check because k has units kg/s².

Amplitude describes maximum displacement from equilibrium, not time per cycle. A larger amplitude increases both the travel distance and the maximum speed proportionally in this linear model, leaving period unchanged. This statement assumes Hooke’s law remains valid and damping is negligible; it is not a license to stretch a real spring arbitrarily.

Squaring the period gives T²=(4π²/k)m. A plot of T² against mass should be linear through the origin for the ideal massless-spring model. A nonzero intercept or curvature can reveal omitted moving spring mass, timing bias, changing stiffness or another assumption failure; it does not by itself identify which mechanism is responsible.

3. Work an example

For m=1 kg and k=20 N/m, ω=√20≈4.472 rad/s, T≈1.405 s and f≈0.712 Hz. Replacing the mass by 4 kg with the same spring doubles T to about 2.810 s. Doubling amplitude within the linear range leaves these periods unchanged.

4. Predict and investigate

If both mass and stiffness double, does the period change?

Choose two masses at one stiffness; predict the period ratio before capture. Then change stiffness to keep m/k fixed. Record T, f and ω separately, including their units. Explain which amplitude claim follows from the model rather than a measured spring.

5. Transfer to a new case

Try a new practice case

A spring oscillator has period 3 s. Stiffness becomes nine times larger at fixed mass. Find its new period.

Your numerical result (s) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Explain the boundary

What additional evidence would separate a moving-spring-mass effect from a timing offset in a nonzero T² intercept?

These are bounded virtual models and supplied numerical cases, not independent measurements or an observed physical investigation. Do not construct spring, pendulum or collision apparatus from these instructions. Practical evidence requires an appropriate supervised setting.

Sources

Original explanations, worked cases and numerical models. Framework checked 30 October 2026: AP Physics 1 Course and Exam Description, topics 7.2; 7.3. Mechanism reference: OpenStax University Physics 1, simple harmonic motion and pendulums. No official assessment items reproduced.