A weight on a string, swinging gently side to side — the simple pendulum is the oldest timekeeper in physics. Give it a small push and each swing takes the same time, no matter how big the push. That time depends on just two things: the length of the string and the strength of gravity. Notice what’s missing — the mass of the bob doesn’t matter at all. Here’s why.
📘 What you need to know
A simple pendulum is a small mass (the bob) on a light, inextensible string fixed at the top
The restoring force is the component of the bob’s weight acting along the arc, back towards equilibrium
The time period is T = 2π√(L/g)
The period depends on the length L and gravitational field strength g — but not on the mass of the bob
The formula only works for small angles (θ < 10°), where the small-angle approximation sin θ ≈ θ holds
Because it depends on g, a pendulum runs at a different rate on the Moon than on Earth
What is a simple pendulum?
A simple pendulum is about as basic as an oscillator gets: a small heavy bob hanging from a light, inextensible string attached to a fixed point above. “Light” means the string’s mass is ignored; “inextensible” means it doesn’t stretch. The bob is treated as a point mass, and it swings from side to side through its lowest point — the equilibrium position.
Where the restoring force comes from
Pull the bob to one side and gravity does the rest. The bob’s weight (mg) points straight down, but we can split it into two parts: one along the string (which the string’s tension cancels) and one along the arc of the swing. That second part is the restoring force — it always points back towards equilibrium.
The weight mg splits into a part along the string and the restoring part mg sin θ along the arc, which pulls the bob back to the middle.
The small-angle approximation
Here’s the catch. The restoring force mg sin θ depends on sin θ, which isn’t quite proportional to the displacement — so strictly, a pendulum isn’t perfect SHM. But for small angles, there’s a lovely shortcut: when θ is small (and measured in radians), sin θ is almost exactly equal to θ.
Small-angle approximation
sin θ ≈ θ (for θ < 10°)
With that approximation, the restoring force becomes proportional to the displacement — the SHM condition — and the neat period formula drops out. The rule of thumb is that the angle of swing must stay below about 10°. Beyond that, the error grows and the formula slowly stops working.
Why does staying under 10° matter? At 8° the difference between sin θ and θ is only about 0.3% — tiny, so the formula is spot-on. Push to 30° and the gap grows fast, so the pendulum no longer keeps perfect SHM time. Keep the swings small and the maths stays clean.
The time period formula
Putting the small-angle SHM together gives the equation for the period of a simple pendulum:
Time period of a simple pendulumT = 2π√(L / g)
Where:
T = time period (s)
L = length of the string, from the pivot to the centre of the bob (m)
g = gravitational field strength (N kg−1, or equivalently m s−2)
The most striking thing about this formula is what isn’t in it: there’s no mass. A heavy bob and a light bob on strings of the same length swing with exactly the same period. That’s the pendulum version of the fact that all objects fall at the same rate under gravity.
WE 1
A simple pendulum has a string of length 0.45 m. Calculate its time period on Earth (g = 9.81 N kg⁻¹).
Step 1 — write down the formulaT = 2π√(L/g)
Step 2 — substitute L = 0.45 m, g = 9.81T = 2π√(0.45 / 9.81) = 2π√(0.0459)Step 3 — evaluateT = 2π × 0.214 = 1.346 sT = 1.35 sDivide L by g first, square-root, then multiply by 2π. The mass never comes into it.
WE 2
What length of pendulum has a time period of exactly 2.0 s on Earth? (g = 9.81 N kg⁻¹)
Step 1 — start from T = 2π√(L/g) and square both sides
T² = 4π² (L/g)
Step 2 — rearrange for L
L = g T² / (4π²)
Step 3 — substitute g = 9.81, T = 2.0 sL = 9.81 × (2.0)² / (4π²) = 39.24 / 39.48L = 0.99 mA pendulum almost exactly 1 metre long ticks once per second each way — that’s the classic “seconds pendulum”.
What changes the period?
🎯 Reading the formula
Longer string → longer period. Since L is on top, a longer pendulum swings more slowly. (Grandfather clocks use long pendulums for slow, steady ticks.)
Stronger gravity → shorter period.g is on the bottom, so more gravity whips the bob back faster.
Mass has no effect. There’s no m in the formula — swapping the bob for a heavier one changes nothing.
Square-root link.T ∝ √L: to double the period you must make the string four times longer.
Gravity matters — the Moon test
Unlike a mass–spring system, a pendulum’s period does depend on g. Take it somewhere with weaker gravity and it swings more slowly. This is a favourite exam comparison.
WE 3
A pendulum of length 0.80 m is taken to the Moon, where g = 1.62 N kg⁻¹. Compare its time period there with its period on Earth (g = 9.81 N kg⁻¹).
On EarthT = 2π√(0.80 / 9.81) = 1.79 sOn the MoonT = 2π√(0.80 / 1.62) = 4.42 sEarth: 1.8 s · Moon: 4.4 sWeaker gravity means a much slower swing — about 2.5× longer, since the period scales with 1/√g.
Longer string L up
T = 2π√(L/g)
Longer period T up
stronger gravity g up → T down
Shorter period
Mass–spring vs simple pendulum
Feature
Simple pendulum
Mass–spring
Period formula
T = 2π√(L/g)
T = 2π√(m/k)
Depends on mass?
No
Yes
Depends on gravity?
Yes
No
Restoring force from
Gravity along the arc
The spring (Hooke’s law)
Valid when
Small angles (< 10°)
Within the spring’s elastic limit
💡 Top tips
T = 2π√(L/g) — length on top, gravity on the bottom, all inside the root.
No mass in the formula — a heavy and a light bob on equal strings swing identically.
To find L, square the equation first: L = gT²/(4π²).
The formula only holds for small swings (< 10°) — the small-angle approximation sin θ ≈ θ.
Measure L from the pivot to the centre of the bob, not to its top.
⚠ Common mistakes
Thinking a heavier bob changes the period — mass has no effect on a pendulum
Using the pendulum formula for large swings — it only works for angles under ~10°
Mixing it up with the mass–spring formula T = 2π√(m/k)
When finding L, forgetting to square the period — use T²
Measuring the string length to the top of the bob instead of its centre of mass
Leaving the calculator in degrees when the small-angle rule needs radians
Quick recap: A simple pendulum swings in SHM for small angles (sin θ ≈ θ, θ < 10°), driven by the restoring force mg sin θ. Its period is T = 2π√(L/g): longer string means a longer period, stronger gravity means a shorter one, and the mass of the bob makes no difference. Rearranged, L = gT²/(4π²).
You’ve now got both SHM systems the IB expects: the mass–spring (depends on mass, not gravity) and the pendulum (depends on gravity, not mass). Next we’ll follow the energy as an oscillator swings — how it trades kinetic for potential energy and back again, while the total stays fixed. That’s the next page: energy changes in SHM.
Pendulums still swinging past you?
Book a free meeting and we’ll work through the period formula, the small-angle rule and past-paper questions together.
