How a pendulum works
When the bob is pulled to one side, gravity pulls it down while the string holds it on a circular arc. The part of gravity that acts along the arc, called the restoring force, always points back toward the lowest point. The bob speeds up as it falls, overshoots the bottom because of its inertia, slows as it climbs the other side, and swings back. This repeating motion is an oscillation.
What sets the period?
Only two things matter much for a pendulum with small swings: the length of the string and thestrength of gravity. A longer string gives a longer period; stronger gravity gives a shorter one. The mass of the bob makes no difference, because a heavier bob needs more force to accelerate and receives exactly that much more force from gravity. The starting angle matters only a little: for swings under about 15°, the period is almost the same whatever the amplitude. Galileo noticed this property, called isochronism, in the early 1600s.
Formulas
For small angles, the period of a simple pendulum is:
T = 2π √(L / g) frequency f = 1 / TThe formula comes from the equation of motion θ'' = −(g/L) sin θ, using the approximation sin θ ≈ θ for small angles. The simulator solves the full equation without this approximation, which is why its measured period grows slightly at large angles. A good correction for moderate angles θ₀ (in radians) is:
T ≈ 2π √(L/g) · (1 + θ₀²/16)Energy in a swing
At the highest points the bob is momentarily still: all its energy is gravitational potential energy(PE = m·g·h). At the lowest point it moves fastest: the energy is all kinetic (KE = ½·m·v²). In between, energy flows back and forth while the total stays constant — unless friction is switched on, in which case the total slowly drains away as heat and the swings shrink.
Running a fair experiment
A scientific investigation changes one independent variable at a time, measures adependent variable (here, the period) and keeps all other control variables the same. Try recording five lengths at the same angle and mass, then plot T against √L: the points form a straight line, confirming the formula.
Key terms
- Period (T)
- Time for one complete oscillation, in seconds.
- Frequency (f)
- Number of oscillations per second, in hertz (Hz). f = 1/T.
- Amplitude
- The maximum displacement from the rest position — here, the starting angle.
- Restoring force
- A force that pulls a system back toward equilibrium; for a pendulum, m·g·sin θ.
- Simple harmonic motion
- Oscillation in which the restoring force is proportional to displacement. A pendulum approximates it for small angles.
- Damping
- Loss of energy (for example to air friction) that makes oscillations die away.
Real-world applications
- Clocks: pendulum clocks, invented by Christiaan Huygens in 1656, were the most accurate timekeepers until the 20th century. A pendulum about 0.99 m long has a period of 2 s — the classic “seconds pendulum”.
- Measuring gravity: because T depends on g, precise pendulums were used to map small variations in Earth’s gravity.
- Foucault’s pendulum: a long, heavy pendulum whose swing plane slowly rotates, demonstrating that Earth turns.
- Buildings and bridges: tuned mass dampers — giant pendulum-like weights — reduce swaying in skyscrapers such as Taipei 101.
- Walking: your legs swing partly like pendulums, which is one reason taller people tend to take slower strides.
Common misconceptions
- “A heavier bob swings faster.” Mass does not change the period.
- “A bigger swing takes much longer.” For small angles, the period hardly changes with amplitude: a bigger swing also moves faster.
- “Friction changes the period a lot.” Light damping shrinks the amplitude but changes the period very little.
What the model simplifies
The model is an ideal simple pendulum: a point mass on a massless, unstretchable string, swinging in one plane. Real pendulums have a bob with size and a string with mass, which slightly lengthen the period. Air friction is modelled as a simple force proportional to speed.
Key takeaways
- For small swings, T = 2π√(L/g): the period depends on length and gravity only.
- Mass has no effect on the period; amplitude has only a small effect at large angles.
- Energy changes between potential and kinetic during each swing.
- Good experiments change one variable at a time and repeat measurements.