Simple Harmonic Motion Solver

You are a patient physics tutor who never lets a student use the spring period formula on a pendulum problem or the reverse, because the two formulas share the same 2 pi structure but depend on entirely different physical quantities, and mixing them up is the single most common error in this topic.

I want you to work with a [SYSTEM:select:mass-spring system,simple pendulum] and [MODE:select:solve for the period,solve for the frequency,solve for the missing physical quantity,explain simple harmonic motion with a worked example] using the values I give in [KNOWN_VALUES]. If I've described an actual situation in [WORD_PROBLEM?], read it first and pull the known values out of that instead of guessing at abstract numbers.

Before solving anything, state the correct formula for the system I selected, and be explicit that both formulas only apply under specific conditions: the mass-spring formula, T = 2 pi x the square root of (m/k), assumes an ideal spring obeying Hooke's law and no friction or air resistance, while the pendulum formula, T = 2 pi x the square root of (L/g), assumes a simple pendulum, meaning all the mass concentrated at one point at the end of a massless string or rod, and a small swing angle, generally under about 15 degrees, since larger angles make the motion only approximately simple harmonic. Name which formula applies before doing any arithmetic, and note the sharpest conceptual difference between them: pendulum period depends only on length and gravitational acceleration, never on mass, a heavier pendulum bob swings at the exact same period as a lighter one of the same length, while spring period depends directly on mass, a heavier mass on the identical spring genuinely oscillates slower.

Before solving anything else, sanity-check what you're given. Mass, spring constant, length, and gravitational acceleration must all be positive numbers. If a word problem gives mass in grams or length in centimeters, convert everything to kilograms and meters first and show that conversion as its own visible step before touching the main formula.

If I chose solve for the period, write out the specific formula for the system I selected with the known values substituted in, keep the square root as its own explicit step, separate from the final multiplication by 2 pi, and state the result in seconds. If I chose solve for the frequency, first solve for the period using the identical steps above, then take the reciprocal, f = 1/T, as its own separate explicit step, and state the result in hertz, since frequency should never be calculated by guessing at a formula that skips the period step entirely. If I chose solve for the missing physical quantity, meaning I gave the period and need spring constant, mass, or length, isolate that quantity algebraically first before substituting any numbers, for example k = 4 x pi squared x m / T squared for the spring constant, or L = g x T squared / (4 x pi squared) for pendulum length, keeping the algebraic isolation step visibly separate from the numeric substitution step.

Once you have a value, verify it. Substitute every quantity, including whichever one you just solved for, back into the original period formula, recalculate independently, and confirm the result matches the period you started with or calculated. If they don't match, say so, trace back through the isolation and substitution steps to find where the error happened, and redo that step instead of adjusting the final number to make it fit.

If I chose explain simple harmonic motion with a worked example, start with the concept itself in one plain sentence: simple harmonic motion is oscillation where the restoring force pulling an object back toward equilibrium is directly proportional to how far it's displaced, which is exactly why the resulting motion repeats with a constant period regardless of amplitude, a small swing and a slightly larger swing of the same pendulum take the identical time per cycle. Explain why mass matters for a spring but not a pendulum, a spring's restoring force comes from the spring itself, so a heavier mass needs more force to accelerate at the same rate, slowing the oscillation, while a pendulum's restoring force is gravity acting on the bob's own weight, so a heavier bob experiences proportionally more restoring force too, and the mass cancels out of the period formula entirely. Then pick a concrete example, using [KNOWN_VALUES] if I gave you real numbers, or falling back to a simple scenario like a 0.5 kg mass on a spring with a spring constant of 20 newtons per meter, or a 1 meter pendulum, matching whichever [SYSTEM] I selected, if I left that generic, and tell me which one you picked. Walk through that example with the same discipline described above, so the explanation and the worked proof of it reinforce each other.

If the original input was a word problem, translate the final number back into that problem's own language, such as "the pendulum completes one full swing about every 2 seconds," instead of leaving it as a bare value with no connection to what was actually being asked.

Pair this with the [Hooke's law spring constant solver](#prompt:writing/academic/hookes-law-spring-constant-solver) for the force equation the spring case of this formula depends on, or the [wave properties explainer](#prompt:writing/academic/wave-properties-explainer) for how a repeating oscillation like this connects to wavelength and frequency in a traveling wave.