T = 2π√(L/g) · the restoring force · why the mass cancels
Hang a marble on a string and then a cannonball on the same string, and they keep time with each other. That result surprises almost everyone, and the reason behind it is the single idea this whole chapter is built on: something pulls it back, harder the further it has gone.
This page covers the restoring force that every oscillation needs, the period and what really controls it, the amplitude that turns out not to, and why a pendulum and a spring obey the same shape of equation. By the end you'll be able to:
Clocks, music, radio, earthquakes and the definition of the second itself all rest on one repeating motion. Learn it here and it never needs learning again:
Searched as: time period of simple pendulum, why mass does not affect pendulum, shm formula, restoring force meaning, amplitude and period.
Predict the period, lock it, then set it swinging. Try the mass slider early — it is the most useful thirty seconds on this page. 🟢 real pendulum engine
Guess, then lock — the bench waits until you commit.
Set a length and a bob, then watch the period — and try changing the bob on its own.
Before any formula, find the restoring force. Six situations — your answers build the table underneath, and one of the six does not oscillate at all.
A pendulum pulled to one side. What pulls it back towards the middle?
Ask one thing: move it away, and is there anything pushing or pulling it back?
Every oscillation on this page has the same two ingredients: something that pulls it home, and enough speed to overshoot.
You are given a target swing time. Choose a string length that delivers it — 3 attempts.
Set the length, then time it. T = 2π√(L/g).
Each claim breaks the formula somewhere. Tap the broken idea.
There is no mass in T = 2π√(L/g). A heavier bob is pulled harder by gravity and is harder to shift by the same factor, so the two cancel exactly.
The length sits under a square root, so twice the length gives √2 ≈ 1.41 times the period, about 2.8 s. To double the period you need four times the string.
Gravity is in the denominator, so weaker gravity means a longer period. On the Moon the same pendulum takes about 2.5 times as long, and the clock loses badly.
Thinking beats memorising. Sort every claim about oscillation into three boxes.
One position in the swing, one decision. Where the speed is greatest is where most marks are lost.
Think about where the height went. Height becomes speed, and speed becomes height again.
Pick the tick you want, then find the string that gives it. Notice there is no bob to choose — because it would not matter.
Rearranged, L = g(T/2π)².
Read the scene. All three are about what sets a rhythm and what does not.
The swing has a period of its own, set by the length of its ropes. Push at that rate and every push adds energy; push at any other rate and half your pushes fight the swing. That matching is what resonance means.
Amplitude and frequency are separate. Plucking harder gives a bigger swing, which is a louder note, but the rate of vibration is fixed by the string's length, tension and thickness. To change the pitch you press a fret and shorten it.
The bridge had a slight sideways sway of its own. Walkers adjusted their steps to match it, which pushed it in time, exactly like the child on the swing. Dampers were fitted to drain that energy and it reopened two years later.
In plain terms: pull something away from where it is happy, and if the pull back grows in step with how far you moved it, you get a swing that repeats with a time of its own.
Move a pendulum aside and the sideways part of its weight pulls it back. Stretch a spring and it pulls back. In both, the further you go the harder the pull, which is what F = −kx says — the minus sign meaning the force points back towards the middle. Anything that behaves this way swings with a sine curve, whatever it is made of.
By the time it reaches the middle the pull back has fallen to nothing — but it is now moving at its fastest. Nothing stops it there, so it carries straight on and starts climbing the other side, and the force turns round to slow it. Fastest where the force is zero, still where the force is largest: those two facts are worth more than any formula on the page.
Double the mass and gravity pulls it back with twice the force — but it is also twice as reluctant to move. The two doublings cancel, and the period stays put. On a spring this does not happen, because the spring's pull does not care about the mass hanging on it, so there a heavier mass really does swing slower.
Past about 20° the small-angle formula starts to under-read, and the real period creeps up by a few per cent. Damping shrinks the amplitude every swing while barely touching the period. And resonance is what happens when you push at exactly the period the thing already has — useful on a swing, dangerous on a bridge.
Exams reward the method. Work it one step at a time — read the thought, predict the line, then reveal it.
Three small labs. The middle one is the one to sit with.
Six wrong ideas, most of them about the bob. Tap a card to bust the myth.
Watching a hanging lamp swing in Pisa cathedral, Galileo timed it against his own pulse and found that wide swings and narrow ones took the same time.
Christiaan Huygens coupled a pendulum to an escapement. Clock error fell from about fifteen minutes a day to about fifteen seconds.
Huygens proved that the period depends on the length and on gravity, and on nothing else — which also turned the pendulum into an instrument for measuring gravity itself.
A vibrating quartz crystal kept far better time than any weight on a string, and the pendulum left the laboratory for good.
The second stopped being a fraction of a day and became 9,192,631,770 oscillations of a caesium atom. The idea is unchanged: count something that repeats.
Its accuracy comes from the period being independent of how wide it swings, so the clock keeps time even as the drive weight runs down.
Length, tension and thickness set the frequency. Plucking harder only changes the amplitude, which your ear hears as loudness.
Every structure has a period of its own. Dampers exist to drain energy before footsteps or wind can feed that period.
One sentence each: what is swinging, and what counts it.
Drag the slider. Six things that repeat, from one swing every two seconds to nine billion a second.
Ten orders of magnitude, and the same question underneath every one: what pulls it back?
Each one is a build and a measurement — the measuring is what turns a demo into a project.
Build: a nut on a thread, a metre rule and a stopwatch.
Measure: the time for twenty swings, then divide by twenty. Your reaction error is divided by twenty as well.
Build: the same length of thread, with bobs from a paperclip to a heavy nut.
Measure: twenty swings for each. Expect the same answer every time, and say so in advance.
Build: one pendulum, adjusted to six different lengths.
Measure: plot T² against L. The gradient is 4π²/g, so the graph hands you g to within a few per cent.
Build: a spring hung from a clamp, with slotted masses.
Measure: the period for each mass, then plot T² against m. Another straight line, with gradient 4π²/k.
Build: a pendulum with a card taped to the bob to catch the air.
Measure: the amplitude every ten swings. It falls by a constant fraction each time, and the period barely moves.
Build: a pendulum whose support you can move by hand at a steady rate.
Measure: the amplitude for several driving rates. It peaks sharply at the pendulum's own period.
Straight answers, each one complete on its own:
Doubling the mass doubles the gravitational pull on the bob, but it also doubles how reluctant the bob is to be accelerated. The two effects cancel exactly, so the acceleration at any point of the swing is unchanged and the timing stays the same. This is the same cancellation that makes all objects fall at the same rate.
It is any force that points back towards the resting position and grows as the object moves further away from it. Gravity does this for a pendulum, a spring's tension does it for a mass on a spring, and the stiffness of the material does it for a twanged ruler. Without one, a displaced object simply stays where it was put.
A longer string means a gentler slope for the same sideways displacement, so the restoring force is weaker, but it also means a longer path to travel. The two effects combine so that the period grows as the square root rather than in proportion. In practice this means four times the length for twice the period.
For small swings, below about ten degrees, it makes essentially none, which is what Galileo noticed and what makes pendulum clocks possible. Beyond about twenty degrees the approximation fails and the real period becomes noticeably longer than the formula predicts, by roughly one per cent at twenty degrees and more beyond that.
The period is the time for one complete cycle, measured in seconds. The frequency is how many cycles happen in one second, measured in hertz. They are simply reciprocals of each other, so a two-second period is half a hertz. Clocks tend to be described by period and sounds by frequency, but they carry the same information.
At the lowest point, where all the height it lost has become speed. That is also where the restoring force is zero, which is why it does not stop there but carries straight on. At the two extremes the opposite holds: the speed is zero for an instant and the restoring force is at its largest.
It would swing, but far too slowly to keep Earth time. Lunar gravity is about 1.6 m/s² against Earth's 9.8, and since g sits in the denominator under a square root, the period becomes about 2.5 times longer. The clock would lose roughly fifteen hours a day unless its pendulum were shortened to about a sixth.
For a pendulum the restoring force comes from gravity, which grows with mass, so the extra mass is cancelled. A spring's pull depends only on how far it is stretched and not at all on what is hanging from it, so adding mass adds reluctance to move without adding any force. That is why T = 2π√(m/k) has the mass in it.
Spot these before the exam does.
Seven question formats, the way LLOS.ai serves them — multiple choice, multiple-correct, fill-in-the-blank, match, sequence, read-think-connect, and write-your-own. Every question has layered hints: a quick nudge, the reasoning, then a deeper connection — so a wrong answer opens a door, never a dead end. 🟢 received from a board-tagged question bank · seed toward 2,000
Pick your board — the set re-tunes to its wording and emphasis. Competitive draws the JEE / NEET / Olympiad lane.
Four pendulums. Tap them in order, slowest swing first. Then read the arithmetic — and see what fooled you.
A five-kilogram iron ball hanging on twenty-five centimetres of steel wire.
A fifty-gram wooden bead on a full metre of cotton thread.
A two-kilogram brass weight on two and a quarter metres of cord.
A two-hundred-gram nut on sixty-four centimetres of string.
Tap the four in the order you believe, slowest swing first.
Galileo watched a lamp and counted his own pulse. Four centuries later we define the second by counting something else that swings — and the question is still the same one: what pulls it back?