Elastic and plastic behaviour
The paperclip test
- Bend a paperclip a little and it springs back — elastic.
- Bend it far and it stays bent — plastic.
- The same material can do both, depending on how hard you push it.
Three stages of stretching
- Elastic and straight — obeys Hooke's law; returns to shape.
- Elastic but curved — past the limit of proportionality; still returns to shape.
- Plastic — past the elastic limit; a permanent change is left.
Match each stage of stretching to what happens.
Elastic = returns to its first length on unloading; plastic = a permanent change once you pass the elastic limit.
Loading and unloading
- Stretch past the elastic limit, then remove the load.
- The unloading line is parallel to the first line but shifted — leaving a permanent extension.
Past the elastic limit, an object returns to its original length when the load is removed.
No — in the plastic region a permanent extension stays after unloading.
Energy = area under the graph
- The work done stretching a material is the area under its force–extension graph.
The energy stored in a stretched material equals the ____ under the force–extension graph.
Work done = area under the force–extension graph (a triangle for a Hooke material).
Stored elastic energy
- For a Hooke's-law material the area is a triangle: $E_{\text{P}} = \tfrac{1}{2}Fx = \tfrac{1}{2}kx^{2}$.
- An equal form is $E_{\text{P}} = \dfrac{F^{2}}{2k}$.
A spring of constant $200\ \dfrac{\text{N}}{\text{m}}$ is stretched by $0.10\ \text{m}$. How much elastic PE is stored?
$E_{\text{P}} = \tfrac{1}{2}kx^{2} = \tfrac{1}{2} \times 200 \times 0.10^{2} = 1.0\ \text{J}$.
When it's not a straight line
- For rubber, or a spring past its limit, the graph is a curve.
- Find the area by counting squares or using trapezia — the rule is the same.
Comparing and releasing
- Same force, softer spring (smaller $k$) → bigger $x$ → more energy stored.
- Released onto a mass, the elastic PE becomes kinetic energy: $\tfrac{1}{2}kx^{2} = \tfrac{1}{2}mv^{2}$.
Two springs are pulled with the same force. The softer spring (smaller $k$) stores:
Same $F$, smaller $k$ → larger extension $x$ → larger area $\tfrac{1}{2}Fx$, so more energy is stored.
When a stretched spring is released onto a mass, its elastic PE becomes ____ energy.
Set $\tfrac{1}{2}kx^{2} = \tfrac{1}{2}mv^{2}$ (plus $mgh$ if it rises) to find the speed.
You've got it
- elastic returns to shape; plastic leaves a permanent extension (past the elastic limit)
- energy stored = area under the force–extension graph
- for a Hooke material $E_{\text{P}} = \tfrac{1}{2}kx^{2}$