Springs and Hooke’s law

Before you start

Put something on a spring balance and the spring stretches; we use that stretch to read a force. This familiar action needs a reliable relationship: the same stretch must correspond to the same force. Our ideal spring uses a simple one. A stretch of 0.2 m gives a restoring force of 2.4 N; 0.4 m gives 4.8 N, and 0.6 m gives 7.2 N. Doubling the stretch doubles the force size. Plotting these pairs gives a straight line, a reusable rule like those on the function page. Real springs follow this relationship approximately over a limited range, not for unlimited stretching. We can now release the object, examine the back-and-forth motion, and compare it with a pendulum.

A stretched spring pulls a block back, while a compressed spring pushes it away. Let us call the position where the spring is neither stretched nor compressed $x = 0$ , and compare the forces on either side. After we release the block, will it stop when it reaches $x = 0$ ? Following its motion will help us connect the spring force with speed and energy.

The setupA block slides on a frictionless horizontal rail and is attached to a spring fixed to the left wall. $x = 0$ marks the natural length of the spring, and positions to the right are positive. The red arrow is the spring force on the block; the cyan arrow is the velocity of the block, not a force. Set the starting stretch or compression and the spring constant $k$ , then press Play to release the block from rest.

What to do and noticeChoose a stretch and keep $k$ as it is. Before pressing Play, read the red arrow: it points back towards $x = 0$ , and doubling the stretch doubles its length. Now release the block and watch it at $x = 0$ . The force there is zero, but the velocity is at its largest, so the block keeps going and compresses the spring on the other side. Finally, keep the same stretch, increase $k$ , and compare the force and how quickly the block returns.

01 / 03 · Use a measured force to find the spring constant

Predict firstAt equal deformation, what changes with a stiffer spring?

Preparing the model…

Guided lesson

Why does a stretched spring pull back, and what does its spring constant mean?

Prerequisites: A block slides on a frictionless horizontal rail, attached to a spring fixed to the left wall.

x = 0

marks its natural length; right is positive. Set the initial stretch or compression, then play to release it from rest.

01 / 03

Use a measured force to find the spring constant

Predict firstAt equal deformation, what changes with a stiffer spring?

First hold the block instead of releasing it. $x$ is signed displacement from the spring’s natural-length position, positive to the right and negative in compression. Experiments show that, within a limited range, doubling the deformation doubles the holding force. We call the ratio $F/x$ the spring constant $k$ and use Hooke’s law for this linear range.

F_{\rm spring}=-kx

The minus sign makes the force point back towards natural length as a restoring force. For $k=12\ \mathrm{N/m}$ and $x=0.6\ \mathrm{m}$ , force is $-12\times 0.6=-7.2\ \mathrm{N}$ , to the left. For $x=-0.6\ \mathrm{m}$ , it is +7.2 N, to the right. Different springs have different $k$ , and the linear rule does not extend beyond the spring’s elastic limit.

How to observe

At equal deformation, what changes with a stiffer spring?
Does the block stop where the spring force is zero?
Where is the energy at the ends and in the middle?

What this model simplifies

Massless Hookean spring within its linear elastic range; no damping or block rotation. Vertical weight and support cancel. Colours indicate elastic potential and kinetic energy, not temperature.

SourcesOpenStax · Mechanics