Simple pendulum — small angle approximation, T = 2π√(L - g) derivation: Foundations

Before you can read the derivation of the parent note (T = 2π√(L/g)), you must be fluent in every symbol it throws at you. This page builds each one from nothing — plain words, then a picture, then why the topic needs it. Read top to bottom; nothing later uses anything not yet defined.

1. The pendulum picture (the stage everything lives on)

Everything below happens on ONE drawing: a string hanging from a fixed point, a ball at the end, and gravity pulling straight down. Let us name the parts.

Figure — Simple pendulum — small angle approximation, T = 2π√(L - g) derivation

WHY start here? Every symbol coming up is a measurement taken off this picture. If the picture is clear, the symbols are just labels.

2. Length $L$ — the size of the swing arc

Look at the amber line in the figure above: that is $L$ . It sets how big the arc is.

WHY the topic needs it: $L$ is the only thing about the pendulum's shape that survives into the final formula $T = 2 π L / g$ . Everything else cancels.

3. The angle $θ$ — how far you have swung

WHY an angle and not a sideways distance? Because the bob is stuck on a string — it can only move along a curved arc, and an angle is the natural way to say "how far along the curve." The sign of $θ$ (positive vs negative) is what lets us later say "the force always points back to $θ = 0$ ."

4. Radians — why $θ$ must be measured this way

Here is a subtle but essential idea the parent note demands: angles measured in radians, not degrees.

Before the formula, one more symbol. The arc length, written $s$ , is the distance the bob travels along its curved path — measured in metres, like walking around the rim of a circle rather than cutting straight across. It is the curved-path cousin of ordinary distance.

WHY the topic needs radians specifically:

The parent's key approximation $sin θ \approx θ$ is only true in radians. In degrees $sin (10°) = 0.174$ but the number " $10$ " is nowhere near it.
The arc-length formula $s = L θ$ (used in Step 2 of the derivation) has NO extra constant only in radians.

5. Mass $m$ and gravity $g$

WHY both $m$ and $g$ appear: gravity's pull is $m g$ , but the bob's reluctance to move is also set by $m$ . The magic of the pendulum (shown in the parent) is that these two $m$ 's cancel — that is why period does not depend on mass.

6. Force, and the minus sign of "restoring"

7. Splitting gravity: the tangent, the arc, and the tension

Two forces pull on the bob: its weight $m g$ (straight down) and the string tension. Let us name the tension.

Since tension lies along the string, it cannot drive motion along the curved path. So we split the weight $m g$ into two pieces: one along the string (which tension balances) and one along the arc (called tangential, meaning "sideways along the curve"). Only the tangential piece drives the swing.

WHY does $sin θ$ appear, and what is $sin$ ?

In our split, the weight $m g$ is the hypotenuse of a little triangle, and the tangential (arc-wise) part is the side "opposite" the angle $θ$ — so it is $m g sin θ$ . The minus sign is the restoring rule from section 6. WHY $sin$ and not $cos$ ? Because at the bottom ( $θ = 0$ ) there should be zero sideways push, and $sin 0 = 0$ — it matches reality; $cos 0 = 1$ would not.

8. Time $t$ , and the rates of change $\frac{d θ}{d t}$ , $\frac{d ^{2} θ}{d t ^{2}}$

Before we can talk about how fast things change, we need the thing they change with: time.

The parent note writes $\frac{d ^{2} θ}{d t ^{2}}$ . Let us earn that notation.

WHY the topic needs a second derivative: Newton's law says force sets acceleration (not position, not speed). Acceleration is the second rate of change of position — hence two $d$ 's. This turns "gravity pulls back" into a solvable equation of motion. More in Simple Harmonic Motion and Angular Frequency and Period.

9. Angular frequency $ω$ and period $T$

WHY the topic needs $ω$ : the tidy SHM equation reads $\frac{d ^{2} θ}{d t ^{2}} = - ω^{2} θ$ . The number sitting in front, $ω^{2}$ , is the physics — for the pendulum it turns out to be $g / L$ , which instantly gives the period. See Angular Frequency and Period.

10. The Taylor idea (just enough to read Step 3)

WHY the topic needs it: with $sin θ$ the equation is unsolvable by hand; with $θ$ it becomes exactly the SHM equation. This single swap is the pivot of the whole derivation. Full story in Taylor Series and Small-Angle Approximations.

Prerequisite map

Equipment checklist

Test yourself — reveal only after answering.

What does

L

measure, and to which point on the bob?

The distance from the pivot to the bob's center of mass, in metres.

What is

θ

, and where is it zero?

The angle of the string from the vertical; zero at the bottom (rest).

Define one radian in one sentence.

The angle whose arc length equals the circle's radius (

\approx 57.3°

What does the symbol

s

mean, its units, and its sign convention?

Signed arc displacement — how far along the curved path the bob is from the bottom, in metres, positive to the right and negative to the left (same sign as

θ

Write arc length in terms of

L

and

θ

, and state the units condition.

s = L θ

, valid only when

θ

is in radians.

What is

sin θ

on a right triangle?

Opposite side over hypotenuse.

Why does the tangential force use

sin

, not

cos

θ = 0

the sideways push must vanish, and

sin 0 = 0

matches that.

What does the minus sign in

- m g sin θ

physically mean?

The force always points back toward equilibrium (restoring).

What is

T_{string}

, and why can't it drive the swing?

The string tension, directed along the string toward the pivot; being along the string it has no component along the arc.

What is

t

, and what depends on it?

Elapsed time in seconds; the angle

θ (t)

changes with it.

What does

\frac{d ^{2} θ}{d t ^{2}}

represent?

Angular acceleration — how fast the angular velocity changes each second of time

t

Relate

T

and

ω

T = 2 π / ω

(one cycle is

2 π

of phase).

State the small-angle approximation and its condition.

sin θ \approx θ

for small

θ

measured in radians.

What are

m

and

g

, with units?

m

= mass (kg, inertia);

g

= gravitational field strength (

\approx 9.8 m/s^{2}

Connections

Simple Pendulum — small angle approximation, T = 2π√(L - g) derivation — the parent this page prepares you for
Restoring Force and Equilibrium — the minus sign explained fully
Taylor Series and Small-Angle Approximations — where $sin θ \approx θ$ comes from
Angular Frequency and Period — $ω$ and $T$ in depth
Simple Harmonic Motion — the pattern the pendulum obeys
Energy in Oscillations — an alternative viewpoint
Mass-Spring System — the same maths with $k$ and $m$
Measuring g with a Pendulum — putting these symbols to work