フックの法則

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.

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.

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.

String vs spring

• A conical pendulum is a mass swung in a horizontal circle on a string.

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