What is the formula for the period of a pendulum?
T = 2π√(L/g), where L is the length from the pivot to the centre of the bob and g is the acceleration due to gravity. A one-metre pendulum on Earth takes about 2.006 seconds per full swing. The formula contains no mass term at all, because gravity pulls harder on a heavier bob in exactly the proportion that makes it harder to accelerate — the two effects cancel completely.
Does the mass of a pendulum bob affect its period?
No. A lead weight and a cork on identical strings swing at identical rates, which is one of the least intuitive results in mechanics. What does affect it is length — and only as a square root, so quartering the length halves the period. Air resistance eventually separates the two in practice, because it slows a light bob far more than a heavy one, but that is drag, not the pendulum law.
What is the formula for a mass on a spring?
T = 2π√(m/k), with m the mass in kilograms and k the spring constant in newtons per metre. A stiffer spring or a lighter mass gives a faster oscillation. Note what is absent here: the amplitude. Pull the mass twice as far and the period does not change at all — it simply travels twice as fast over twice the distance.
Why doesn't amplitude affect the period in simple harmonic motion?
Because the restoring force grows in proportion to the displacement. Pull the mass twice as far and the spring pulls back twice as hard, so it accelerates twice as fast and covers the doubled distance in the same time. This property is called isochronism, and it is the entire reason a pendulum can keep time — a clock whose swing decays slightly would otherwise run fast or slow as its mainspring unwound.
What is the small-angle approximation, and when does it break?
T = 2π√(L/g) is derived by replacing sin θ with θ, which is only accurate for small swings. The error is 0.19% at 10°, about 0.8% at 20°, 1.7% at 30% and roughly 18% at 90°. For a laboratory measurement with a stopwatch, anything under about 15° is safe. Beyond that the true period is genuinely longer, and this calculator shows both figures so you can see the gap.
How are position, velocity and acceleration related in SHM?
They are the same wave, each shifted a quarter of a cycle from the last. Position is A·cos(ωt), velocity is −Aω·sin(ωt), acceleration is −Aω²·cos(ωt). So velocity peaks exactly where position crosses zero — at the centre of the swing — and acceleration is always the exact mirror of position, largest at the extremes where the object is momentarily still. That last relation, a = −ω²x, is the definition of simple harmonic motion.
What is angular frequency, and how is it different from frequency?
Frequency f counts cycles per second, in hertz. Angular frequency ω counts radians per second, and ω = 2πf. The angular version is what appears inside the sine and cosine, because those functions take radians. For a pendulum ω = √(g/L); for a spring ω = √(k/m). Everything else — period, maximum speed, maximum acceleration — is built from ω.
Where does the energy go during an oscillation?
Nowhere — it swaps. At the extremes the object is momentarily still and all the energy is stored, as spring tension or as the height the bob has risen. At the centre the store is empty and all of it is kinetic. The total stays at ½kA² for a spring, and the exchange happens twice per cycle, not once, because the object passes through the centre twice. Doubling the amplitude quadruples the energy.
What is a seconds pendulum?
One that takes exactly one second per half-swing, so two seconds for a complete cycle. That requires a length of about 0.994 m at sea level, which is why longcase (grandfather) clocks are the height they are — the case exists to house that specific pendulum. Because g varies slightly with latitude and altitude, the exact length differs from place to place, a fact eighteenth-century surveyors used to measure the shape of the Earth.