Simple pendulum
A real pendulum simulation compared with the textbook formula. At what angle does the approximation stop working?
The period of a pendulum
A simple pendulum is a point mass hanging from a massless string of length L. When it is pulled aside by an angle θ, the component of its weight tangent to the path pushes it back down. Applying Newton’s second law gives the equation of motion:
θ'' = −(g / L) · sin θ
This equation has no simple solution. The textbook trick is to assume the angle is small, so that sin θ ≈ θ (in radians). The equation then becomes that of a harmonic oscillator, and its period is the well-known formula:
T₀ = 2π · √(L / g)
Notice what does not appear: neither the mass nor the amplitude. Galileo had already noticed that a pendulum takes the same time to swing whether it moves a lot or a little (isochronism), which is why it was used in clocks for centuries.
When the approximation fails
The simulator does not use the approximation: it solves the full equation numerically (fourth-order Runge-Kutta method). The gray ghost pendulum follows the linear formula. At 10° the two move almost together (0.2% error), at 45° the real pendulum is already 4% slower, and beyond 90° the difference is huge. Near 180° the pendulum almost balances on the inverted vertical and the period tends to infinity.
The exact period can be computed with the arithmetic-geometric mean (AGM): T = T₀ / AGM(1, cos(θ₀/2)), which is what the “Actual period” card shows.
Energy and friction
The bars show how energy turns from potential (at the very top) into kinetic (at the lowest point) and back. Without friction the total is conserved. If you add friction, the energy is dissipated and the amplitude decays exponentially; notice that the period hardly changes as long as the damping is small.
Frequently asked questions
What does the period of a pendulum depend on?
For small swings, only on the length of the string and on gravity: T = 2π · √(L / g). It does not depend on the mass or, as long as the angle is small, on the amplitude. To double the period you have to make the string four times longer.
Why does the period not depend on the mass?
Because the mass appears twice and cancels out: a heavier bob feels a stronger pull of gravity, but it is also harder to accelerate, in exactly the same proportion. It is the same reason why, without air, all objects fall equally fast.
How long is a seconds pendulum?
A pendulum that takes one second to swing from one side to the other has a period of 2 s. Solving the formula, L = g · T² / (4π²) ≈ 0.994 m on Earth: almost exactly one metre. It is the pendulum found in many grandfather clocks.
How do you measure gravity with a pendulum?
Measure the string length up to the centre of the bob and time 10 or 20 full swings at a small angle to reduce the error. Divide the time by the number of swings to get T, then compute g = 4π² · L / T². With a metre of string and a stopwatch you get 9.8 m/s² with less than 1 % error.