Mass on a Spring

Hooke's Law

the spring pulls back

A spring pushes back toward its resting position with a force proportional to how far it is stretched or squashed.

Example

A spring pushes back toward its resting position with a force proportional to how far it is stretched or squashed. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The spring pulls back in proportion to the stretch

Stretch or squash a spring and it pushes back toward its resting position. The restoring force is the stiffness times the displacement, and the minus sign just means it always points back toward the middle. Stretch twice as far and it pulls twice as hard.

F=−k xF = -k\,x
A stretched spring pulls the block backA block pulled to the right, stretching the spring, with a restoring force arrow pointing back toward the middle.mF

A worked value

With stiffness 2 newtons per metre, pulling the block out to 3 metres gives a restoring force of 2 times 3, or 6 newtons, pointing back toward the middle.

F=k x=2 N/m ⋅ 3 m=6 NF = k\,x = 2\ \text{N/m} \,\cdot\, 3\ \text{m} = \hl{6}\ \text{N}

Same spring, more stretch, more restoring force

Hold stiffness fixed. Stretch farther and the spring pulls back harder in direct proportion to displacement.

kxF2 N/m1 m2 N2 N/m2 m4 N2 N/m3 m6 N\begin{array}{c|c|c}k & x & F \\ \hline 2\ \text{N/m} & 1\ \text{m} & 2\ \text{N} \\ 2\ \text{N/m} & 2\ \text{m} & 4\ \text{N} \\ 2\ \text{N/m} & 3\ \text{m} & 6\ \text{N}\end{array}

Same stretch, stiffer spring, more restoring force

Hold stretch fixed. A stiffer spring gives a larger restoring force for the same displacement.

kxF1 N/m3 m3 N2 N/m3 m6 N3 N/m3 m9 N\begin{array}{c|c|c}k & x & F \\ \hline 1\ \text{N/m} & 3\ \text{m} & 3\ \text{N} \\ 2\ \text{N/m} & 3\ \text{m} & 6\ \text{N} \\ 3\ \text{N/m} & 3\ \text{m} & 9\ \text{N}\end{array}
mechanics A 2 N/m spring pulled to 3 m gives a clean 6 N restoring force.