The Pendulum

Pendulum Restoring Force

gravity pulls it back

Gravity on a displaced bob splits, like on a ramp, into a part along the arc that pulls it back and a part along the string the tension balances.

Example

Gravity on a displaced bob splits, like on a ramp, into a part along the arc that pulls it back toward the bottom and a part along the string that the tension balances. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A bob pulled to one side

A 1 kilogram bob hangs from a string and is pulled to one side. Gravity pulls it straight down with weight 1 times 10, or 10 newtons.

W=m g=1 kg ⋅ 10 m/s2=10 NW = m\,g = 1\ \text{kg} \,\cdot\, 10\ \text{m}/\text{s}^{2} = \hl{10}\ \text{N}
A pendulum bob and its weightA bob on a string from a fixed support, pulled to the right, with a weight arrow pointing straight down.bobW

Split the weight along and across the string

The string can only pull along its length, so split the weight like on a ramp, using the right triangle the string makes with the vertical. The across-the-arc part is the weight times 3 over 5, a 6 newton pull back toward the bottom, and the along-string part is the weight times 4 over 5, 8 newtons that the string tension balances.

Frestore=Wsin⁡θ=10 N⋅35=6 NF_{\text{restore}} = W\sin\theta = 10\ \text{N} \cdot \frac{3}{5} = \hl{6}\ \text{N}
The weight split along the arc and along the stringThe bob's weight arrow with its restoring part along the arc and its part along the string.bobWrestoringalong string

Same bob, wider swing, larger restoring pull

Hold the bob and gravity fixed. As the sine of the angle grows, the restoring part of the same weight grows in direct proportion. These are training angles chosen for exact arithmetic; only the small ones belong to the simple-harmonic approximation.

Wsin⁡θFrestore10 N152 N10 N254 N10 N356 N\begin{array}{c|c|c}W & \sin\theta & F_{\text{restore}} \\ \hline 10\ \text{N} & \tfrac{1}{5} & 2\ \text{N} \\ 10\ \text{N} & \tfrac{2}{5} & 4\ \text{N} \\ 10\ \text{N} & \tfrac{3}{5} & 6\ \text{N}\end{array}

Same angle, heavier bob, larger restoring pull

Now hold the angle fixed. A heavier bob has a larger weight, so the same fraction of that weight gives a larger restoring pull.

mWFrestore1 kg10 N6 N2 kg20 N12 N3 kg30 N18 N\begin{array}{c|c|c}m & W & F_{\text{restore}} \\ \hline 1\ \text{kg} & 10\ \text{N} & 6\ \text{N} \\ 2\ \text{kg} & 20\ \text{N} & 12\ \text{N} \\ 3\ \text{kg} & 30\ \text{N} & 18\ \text{N}\end{array}
mechanics A 3-4-5 geometry gives an exact 6 N restoring pull from a 10 N weight.