Hooke's law, stress and strain · 胡克定律、应力与应变
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| Hooke's law/hʊks lɔː/ | 胡克定律 | hú kè dìng lǜ |
| tensile/ˈtensaɪl/ | 拉伸 | lā shēn |
| extension/ekˈstenʃn/ | 伸长量 | shēn cháng liàng |
| compressive/kəmˈpresɪv/ | 压缩 | yā suō |
| load/ləʊd/ | 负载 | fù zài |
| spring constant/sprɪŋ ˈkɒnstənt/ | 劲度系数 | jìn dù xì shù |
| limit of proportionality/ˈlɪmɪt ɒv prəˌpɔːʃəˈnælɪti/ | 比例极限 | bǐ lì jí xiàn |
| stress/stres/ | 应力 | yìnglì |
| strain/streɪn/ | 应变 | yìng biàn |
| Young modulus/jʌŋ ˈmɒdjʊləs/ | 杨氏模量 | yáng shì mó liàng |
Pull twice as hard
- Pull a spring twice as hard and it stretches twice as far.
- That neat straight-line rule is Hooke's law 胡克定律 — true up to a point.
- It lets us measure forces with springs and test materials.
用两倍的力拉
- 用两倍的力拉弹簧,它就伸长两倍。
- 这条整齐的直线规律就是 胡克定律(Hooke's law)——在一定范围内成立。
- 它让我们能用弹簧测量力,并测试材料。
Stretching and squeezing
- A tensile 拉伸 force stretches an object, giving an extension 伸长量 $x$.
- A compressive 压缩 force squeezes it. The applied force is the load 负载.
A modern universal testing machine stretches a sample and records the force and extension
拉伸与压缩
- 拉力(tensile force) 把物体拉长,产生 伸长量(extension) $x$。
- 压力(compressive force) 把它压短。施加的力叫 载荷(load)。
一台现代万能试验机拉伸样品并记录力和伸长量
Hooke's law · 胡克定律
F = k·x
Force is proportional to extension — the gradient is the spring constant k. · 力与伸长量 成正比——斜率是弹簧常数 k。
Hooke's law
- The extension is proportional to the load: $F = kx$.
- $k$ is the spring constant 劲度系数 (stiffness), in $\dfrac{\text{N}}{\text{m}}$ — the gradient of an $F$–$x$ graph.
胡克定律
- 伸长量与载荷成正比:$F = kx$。
- $k$ 是 弹簧常数(spring constant)(劲度),单位 $\dfrac{\text{N}}{\text{m}}$——是 $F$–$x$ 图的斜率。
Hooke's law spring · 胡克定律弹簧
Hang a load on the real spring: up to the elastic limit the extension is proportional to the force; beyond it the spring is stretched for good. · 在真实弹簧上悬挂负载：在弹性限度内，伸长量与力成正比；超过限度后，弹簧会发生永久形变。
A load of $12\ \text{N}$ stretches a spring by $0.040\ \text{m}$. What is the spring constant? · $12\ \text{N}$ 的载荷把弹簧拉伸 $0.040\ \text{m}$。弹簧常数是多少?
$k = \dfrac{F}{x} = \dfrac{12}{0.040} = 300\ \dfrac{\text{N}}{\text{m}}$. · $k = \dfrac{F}{x} = \dfrac{12}{0.040} = 300\ \dfrac{\text{N}}{\text{m}}$。
Within Hooke's law, doubling the load doubles the extension. · 在胡克定律范围内,载荷加倍,伸长量也加倍。
Yes — $F = kx$ is a straight-line (proportional) relationship, so $x$ scales with $F$. · 是的——$F = kx$ 是一种直线(正比)关系,所以 $x$ 随 $F$ 成比例变化。
The limit of proportionality 比例极限
- Hooke's law only holds up to the limit of proportionality.
- Past it the $F$–$x$ line curves — the simple $F = kx$ no longer works.
A spring obeys Hooke's law: extension is proportional to the force applied
比例极限
- 胡克定律只在 比例极限(limit of proportionality) 以内成立。
- 超过它,$F$–$x$ 线就弯曲——简单的 $F = kx$ 不再适用。
一根弹簧服从胡克定律:伸长量与施加的力成正比
The straight-line law $F = kx$ holds only up to the: · 直线规律 $F = kx$ 只在以下哪点以内成立:
Beyond the limit of proportionality the $F$–$x$ graph curves and $F = kx$ no longer applies. · 超过比例极限,$F$–$x$ 图就弯曲,$F = kx$ 不再适用。
A complete statement of Hooke's law contains which of these? Select all · 所有 that apply. · 胡克定律的完整表述包含下列哪些?选出所有适用的。
The condition is part of the law, not an extra. Rubber obeys nothing like it, and a metal stops obeying it beyond the straight line. · 那个条件是定律的一部分,不是额外的补充。橡胶完全不遵守它,而金属过了直线段也不再遵守。
Springs together
- Series (end to end): $\dfrac{1}{k_{\text{total}}} = \dfrac{1}{k_1} + \dfrac{1}{k_2}$ — softer overall.
- Parallel (side by side): $k_{\text{total}} = k_1 + k_2$ — stiffer overall.
Springs in series are softer (1/k adds); springs in parallel are stiffer (k adds)
弹簧的组合
- 串联(series)(首尾相接):$\dfrac{1}{k_{\text{total}}} = \dfrac{1}{k_1} + \dfrac{1}{k_2}$——整体更软。
- 并联(parallel)(并排):$k_{\text{total}} = k_1 + k_2$——整体更硬。
弹簧串联时更软(1/k 相加);并联时更硬(k 相加)
Two identical springs (each constant $k$) are joined in series. The combined spring constant is: · 两根相同的弹簧(各自常数为 $k$)串联起来。合并后的弹簧常数是:
$\dfrac{1}{k_{\text{total}}} = \dfrac{1}{k} + \dfrac{1}{k} = \dfrac{2}{k}$, so $k_{\text{total}} = \tfrac{1}{2}k$ — the pair stretches more easily. · $\dfrac{1}{k_{\text{total}}} = \dfrac{1}{k} + \dfrac{1}{k} = \dfrac{2}{k}$,所以 $k_{\text{total}} = \tfrac{1}{2}k$——这一对更容易被拉伸。
Stress 应力 and strain 应变
- Stress $\sigma = \dfrac{F}{A}$ (force per area, in Pa).
- Strain $\varepsilon = \dfrac{x}{L_0}$ (extension ÷ original length — no unit).
应力与应变
- 应力(stress) $\sigma = \dfrac{F}{A}$(单位面积上的力,单位 Pa)。
- 应变(strain) $\varepsilon = \dfrac{x}{L_0}$(伸长量 ÷ 原长——无单位)。
Match each quantity to its formula. · 把每个量与它的公式配对。
Stress is force per area; strain is the fractional extension; the spring constant is load per extension. · 应力是单位面积上的力;应变是相对伸长量;弹簧常数是单位伸长量上的载荷。
Match each term to the definition the examiner marks. · 把每个术语与评分认可的定义配对。
The Young modulus is a property of the MATERIAL and the spring constant of the SPECIMEN, which is why a thicker wire of the same metal changes one and not the other. · 杨氏模量是材料的属性,劲度系数是样品的属性,所以同种金属的更粗导线只改变其中一个。
The Young modulus 杨氏模量
- $E = \dfrac{\sigma}{\varepsilon} = \dfrac{F L_0}{A x}$, in Pa (about $2 \times 10^{11}$ for steel).
- It is a property of the material only; the spring constant $k = \dfrac{EA}{L_0}$ also depends on size.
杨氏模量
- $E = \dfrac{\sigma}{\varepsilon} = \dfrac{F L_0}{A x}$,单位 Pa(钢约为 $2 \times 10^{11}$)。
- 它只是 材料 的性质;弹簧常数 $k = \dfrac{EA}{L_0}$ 还取决于尺寸。
A material with Young modulus $2.0 \times 10^{11}\ \text{Pa}$ is under a stress of $1.0 \times 10^{8}\ \text{Pa}$. What is the strain? · 一种杨氏模量为 $2.0 \times 10^{11}\ \text{Pa}$ 的材料受到 $1.0 \times 10^{8}\ \text{Pa}$ 的应力。应变是多少?
$\varepsilon = \dfrac{\sigma}{E} = \dfrac{1.0 \times 10^{8}}{2.0 \times 10^{11}} = 5.0 \times 10^{-4}$. · $\varepsilon = \dfrac{\sigma}{E} = \dfrac{1.0 \times 10^{8}}{2.0 \times 10^{11}} = 5.0 \times 10^{-4}$。
The Young modulus of a wire depends on its length and thickness. · 金属丝的杨氏模量取决于它的长度和粗细。
No — the Young modulus is a property of the material. The spring constant $k = EA/L_0$ is what depends on size. · 不——杨氏模量是 材料 的性质。取决于尺寸的是弹簧常数 $k = EA/L_0$。
A wire has diameter 0.50 mm. What is its cross-sectional area, in units of 1e-7 m^2? · 一根导线直径 0.50 mm。它的横截面积是多少(以 1e-7 m^2 为单位)?
Radius 0.25 mm = 2.5e-4 m, so A = pi r^2 = 1.96e-7 m^2. Using the diameter as the radius gives four times too much, the classic error here. · 半径 0.25 mm = 2.5e-4 m,所以 A = pi r^2 = 1.96e-7 m^2。把直径当半径会大出四倍,这是这里的经典错误。
Measuring the Young modulus
- Use a long, thin wire so the extension is big enough to measure.
- Find the area from the diameter (a micrometer), load it step by step, and plot $F$ against $x$: gradient $= \dfrac{EA}{L_0}$.
A long, thin wire is loaded step by step; the gradient of F against x gives EA/L₀
测量杨氏模量
- 用一根 又长又细 的金属丝,使伸长量大到足以测量。
- 用直径(千分尺)求面积,逐步加载,并画 $F$ 对 $x$ 的图:斜率 $= \dfrac{EA}{L_0}$。
一根又长又细的金属丝逐步加载;F 对 x 图的斜率给出 EA/L₀
A thicker wire of the same metal has a larger Young modulus. · 同种金属更粗的导线有更大的杨氏模量。
The Young modulus is the material's own property, unchanged by thickness. The thicker wire has a larger spring constant, which is the specimen's property. · 杨氏模量是材料自身的属性,不随粗细改变。更粗的导线劲度系数更大,而那是样品的属性。
Marks that slip away
- Hooke's law needs its condition. "The extension is directly proportional to the applied force" is only half of it; add "provided the limit of proportionality is not exceeded".
- The limit of proportionality is the end of the straight line. The elastic limit is the end of elastic behaviour, a little beyond it. They are not the same point.
- In $A = \pi r^2$ use the radius, so halve a diameter first, and convert $\text{mm}^2$ to $\text{m}^2$ by $10^{-6}$, not $10^{-3}$.
- A thicker wire does not have a larger Young modulus. The modulus belongs to the material; the thicker wire has a larger spring constant.
- Stress is force per unit cross-sectional area and strain is extension per unit original length. Strain has no unit.
容易丢掉的分
- 胡克定律必须带条件。"伸长量与所加的力成正比"只是一半;还要加上"在不超过比例极限的前提下"。
- 比例极限是直线段的终点。弹性极限是弹性行为的终点,略在它之后。两者不是同一点。
- $A = \pi r^2$ 里用的是半径,所以直径要先减半,而 $\text{mm}^2$ 换 $\text{m}^2$ 要乘 $10^{-6}$,不是 $10^{-3}$。
- 更粗的导线的杨氏模量并不更大。模量属于材料;更粗的导线只是劲度系数更大。
- 应力是单位横截面积上的力,应变是单位原长上的伸长。应变没有单位。
You've got it
- Hooke's law $F = kx$ holds up to the limit of proportionality
- stress $\sigma = \dfrac{F}{A}$, strain $\varepsilon = \dfrac{x}{L_0}$
- Young modulus $E = \dfrac{\sigma}{\varepsilon}$ — a property of the material, not its shape
你掌握了
- 胡克定律 $F = kx$ 在 比例极限 以内成立
- 应力 $\sigma = \dfrac{F}{A}$,应变 $\varepsilon = \dfrac{x}{L_0}$
- 杨氏模量 $E = \dfrac{\sigma}{\varepsilon}$——是材料的性质,与形状无关