Simple Harmonic Motion Calculator
Period and frequency for a mass on a spring or a simple pendulum, with the two absences stated.
What is Simple Harmonic Motion?
Simple harmonic motion is any oscillation whose restoring force is proportional to displacement, and the two textbook oscillators share the mathematics: a mass on a spring with T = 2π√(m/k) and a simple pendulum with T = 2π√(L/g). The interesting physics is in what the formulas leave out — the pendulum's period ignores its mass entirely, and both periods ignore their amplitude, which is why a grandfather clock keeps time whether the weights are heavy and the swing is wide.
How it works
A mode selector picks the oscillator; each takes its own pair — mass and spring constant, or length and g — and computes the period, then derives f = 1/T and ω = √(k/m) or √(g/L) from the shared relation. The working shows the substituted root and names the two absences where the numbers are. The pendulum result carries the small-angle caveat: past about 15° the real period starts to grow, and no formula on the page fixes that.
- Enter what you know. Every field recalculates as you type — there is no submit button. Leave the quantity you want blank and the page solves for it.
- Read the answer and the working. The result appears above the fold; underneath it is the formula with your numbers substituted, so the arithmetic can be checked rather than trusted.
- Check the units and the size. Units are shown on every value and selectable per field, so a result in kJ, kPa or km/h is converted rather than assumed. Sanity-check the magnitude against a known landmark such as 1 g/cm³ for water.
Examples
A textbook problem with a known answer
Every tool here is checked against standard example problems — the molar mass of C6H12O6 is 180.16 g/mol, and free fall from 20 m takes about 2.02 s at g = 9.8 m/s².
A result you want to reason backwards from
Leave the quantity you already know blank instead: given the final answer, most tools recover the starting value — the mass needed for a target concentration, or the height a fall must have had.
Common mistakes
Forgetting that a formula is only as good as its subscripts
H2O and H2O2 are different compounds. The formula field shows its parse — element by element — so a mis-typed subscript is visible before it reaches the arithmetic.
Mixing units in the middle of a calculation
The most common source of a wrong answer is a value in grams next to a formula expecting kilograms, or Celsius next to a gas constant expecting kelvin. This category converts on the way in and labels what it used.
Frequently asked questions
Why doesn't the pendulum's period depend on its mass?
Because gravity pulls harder on a heavier bob in exact proportion to its inertia: the m cancels between the two roles. The same cancellation is why all objects fall at the same rate — the pendulum is free fall constrained to a string.
Why doesn't the period depend on amplitude?
For small swings, a wider pull also means a larger restoring force in exact proportion, so the timing cancels the distance. Past about 15° the proportionality breaks and the period grows — the small-angle approximation is the model's boundary, and the page states it rather than hiding it.
Spring or pendulum — where does g appear?
Only in the pendulum. A spring's timing comes from its own k and the mass; take the same spring to the Moon and it keeps time unchanged, while the pendulum slows. That contrast is a favourite exam question and the mode selector is built to make it visible.