Hooke's law
Introduced| English | Meaning |
|---|---|
| elastic/ɪˈlæstɪk/ | Describing a material or object that returns to its original shape or size after a deforming force is removed, within its elastic limit. |
| Hooke's law/hʊks lɔː/ | The rule that extension is proportional to applied force while a spring remains within its limit of proportionality. |
| extension/ekˈstenʃn/ | The increase in length of an object. |
| tension/ˈtenʃn/ | A pulling force transmitted through a stretched string, cable or similar object. |
| natural length/ˈnætʃərəl leŋθ/ | natural length |
| modulus of elasticity/ˈmɒdjʊləs ɒv ɪlæˈstɪsɪti/ | In the ideal elastic-string model, the parameter lambda in tension equals lambda times extension divided by natural length; it has units of force. |
| elastic potential energy/ɪˈlæstɪk pəˈtenʃl ˈenədʒi/ | Energy stored when an elastic object is stretched or compressed. |
| compress/kəmˈpres/ | To reduce the volume of a substance by applying pressure. |
The physics of a bungee jump
- Stretch an elastic 弹性 rope and it pulls back — harder the more you stretch it.
- That simple proportional law, Hooke's law 胡克定律, governs springs, bungee cords, and the energy they store and release.
An elastic string of natural length 2 m and modulus 50 N is stretched by 0.5 m. Find the tension T = λx/L (N).
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T = 50 × 0.5 / 2 = 12.5 N.
By Hooke's law, the tension is proportional to the extension.
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T = (λ/L)x, so T ∝ x while the material stays elastic.
Hooke's law
- The tension 张力 in an elastic string or spring is proportional to its extension 伸长量 $x$:
- $L$ is the natural length 自然长度 and $\lambda$ the modulus of elasticity 弹性模量 (a property of the material).
Tension rises linearly with extension; the area under the line is the stored elastic energy.
Hooke's law
F = k·x
Force is proportional to extension — the gradient is the stiffness k.
For the same string (λ = 50, x = 0.5, L = 2), find the stored elastic energy E = λx²/(2L) (J).
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E = 50 × 0.25 / 4 = 12.5/4 = 3.125 J.
The elastic energy equals the ______ under the force–extension graph.
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Work done stretching = area under the T–x line = ½ × x × (λx/L) = λx²/(2L).
Elastic potential energy 弹性势能
- Stretching does work, stored as elastic potential energy — the triangular area under the graph:
Worked example
- An elastic string of natural length $2$ m and modulus $50$ N is stretched by $0.5$ m.
- Tension $T = \dfrac{\lambda x}{L} = \dfrac{50 \times 0.5}{2} = 12.5$ N.
- Stored energy $E = \dfrac{\lambda x^2}{2L} = \dfrac{50 \times 0.25}{4} = 3.125$ J.
An elastic string (not a spring):
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A string only pulls; when slack its tension is zero. Only a spring pushes when compressed.
String vs spring
A string can't push. An elastic string only pulls (tension when stretched, nothing when slack). A spring also pushes when compressed 压缩 — so a spring's extension $x$ can be negative.
- A conical pendulum is a mass swung in a horizontal circle on a string.
You've got it
- Hooke's law: $T = \dfrac{\lambda x}{L}$ (tension ∝ extension)
- stored elastic PE: $E = \dfrac{\lambda x^2}{2L}$ (area under the force–extension line)
- $\lambda$ = modulus, $L$ = natural length; a string only pulls, a spring also pushes