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.

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
mechanics A 3-4-5 geometry gives an exact 6 N restoring pull from a 10 N weight.