15.1.1 Elastic and inelastic distortion

Distortion requires opposing forces

1. A single force accelerates an unsupported object, whereas distortion requires forces acting at different points or in different directions.
2. Pulling both ends of a spring stretches it, pushing both ends compresses it, and applying separated forces can bend an object.
3. The forces do not have to be equal while the shape is changing, but both are needed to produce stretching, compression or bending.

Elastic and inelastic changes differ

1. After elastic distortion, the object returns to its original length and shape when the forces are removed.
2. After inelastic distortion, some or all of the change remains because the object has been permanently deformed.
3. A spring can behave elastically for small forces but become inelastically distorted if it is overloaded.

15.1.2 The spring constant

Spring constant measures stiffness

1. For a spring in its linear elastic region, force and extension are related by $F=kx$.
2. In this equation, $F$ is the force exerted on the spring in newtons, $k$ is the spring constant in newtons per metre, and $x$ is the extension in metres.
3. Extension is the change in length, so $x=stretched\; length-original\; length$.
4. A larger value of $k$ means a stiffer spring because more force is required for the same extension.

Rearrange and convert before substituting

1. Use $k=\frac{F}{x}$ to calculate spring constant and $x=\frac{F}{k}$ to calculate extension.
2. Convert centimetres or millimetres to metres before using $F=kx$: $1cm=0.01m$ and $1mm=0.001m$.
3. The unit follows from $k=F/x$, giving ${N\; m}^{-1}$ or $N/m$.

Direct proportion has a precise meaning

1. While $k$ is constant, doubling the force doubles the extension, so $F\propto x$.
2. A force against extension graph is a straight line through the origin in this region, and its gradient equals $k$.

15.1.3 Work done in stretching a spring

Stretching transfers energy to a spring

1. Work done by a stretching force transfers energy to the spring's elastic potential energy store.
2. For a linear spring stretched from zero extension, $E=\frac{1}{2}k{x}^2$.
3. Here, $E$ is the energy transferred in joules, $k$ is the spring constant in newtons per metre, and $x$ is the extension in metres.
4. The equation applies only while the force-extension relationship is linear.

Energy rises with extension squared

1. Because $E\propto x^2$ for constant $k$, doubling the extension makes the stored energy four times larger.
2. The factor $\frac{1}{2}$ appears because the force increases uniformly from zero to $F$ as the spring is stretched.
3. The energy transferred also equals the area beneath a force-extension graph; for a straight line this area is the triangle $\frac{1}{2}Fx$.

Rearrangement links energy to stiffness

1. To find extension, use $x=\sqrt{\frac{2E}{k}}$.
2. To find spring constant, use $k=\frac{2E}{x^2}$.
3. Square the extension in metres, not the centimetre value.

15.1.4 Linear and non-linear force–extension

Graph shape distinguishes linear behaviour

1. A linear force-extension relationship produces a straight line, while a non-linear relationship produces a curve.
2. Direct proportionality requires a straight line through the origin, not merely any straight line.
3. Below the limit of proportionality, $F=kx$ and the gradient of a force against extension graph is the constant $k$.
4. Beyond this limit, extension no longer increases in direct proportion to force and the gradient changes.

Loading and unloading reveal permanent change

1. If unloading returns the graph to the origin, the object has recovered its original length.
2. If the unloading path finishes at a non-zero extension, the object has undergone inelastic distortion.
3. Rubber bands commonly give curved force-extension graphs even when they return to their original length, so non-linear does not automatically mean inelastic.