Spring Forces · 弹簧力
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| Hooke's law/hʊks lɔː/ | 胡克定律 | hú kè dìng lǜ |
| displacement/dɪˈspleɪsmənt/ | 位移 | wèi yí |
| spring constant/sprɪŋ ˈkɒnstənt/ | 弹簧劲度系数 | tán huáng jìn dù xì shù |
| restoring force/rɪˈstɔːrɪŋ fɔːs/ | 回复力 | huí fù lì |
| in parallel/ɪn ˈpærəlel/ | 并联 | bìng lián |
| in series/ɪn ˈsɪəriːz/ | 串联 | chuàn lián |
The more you stretch, the harder it pulls
- Pull a spring a little and it tugs back gently; pull it twice as far and it tugs back twice as hard.
- Let go and it snaps toward its rest position every time.
- This tidy, proportional push-back is what makes springs so predictable.
- It is also the recipe for every oscillation in physics.
你拉得越远,它拉得越紧
- 稍微拉一下弹簧,它轻轻回拉;拉远一倍,它就回拉两倍那么紧。
- 一松手,它每次都弹回到静止位置。
- 这种整齐的、成正比的回推,正是弹簧如此可预测的原因。
- 它也是物理中每一种振动的配方。
Hooke's law
- Hooke's law 胡克定律 describes an ideal spring:
- The force is proportional to how far the spring is stretched or squeezed.
- The minus sign says the force always points back toward equilibrium.
胡克定律
- 胡克定律描述理想弹簧:
- 力与弹簧被拉伸或压缩的距离成正比。
- 负号表示力总是指回平衡位置。
Select all · 所有 true statements about an ideal spring. · 选出关于理想弹簧的所有正确说法。
The first two are Hooke's law. The third is backwards -- a stiffer spring has a larger · 变大 $k$. · 前两条是胡克定律。第三条说反了——更硬的弹簧 $k$ 更大。
Stiffness and displacement
- The spring constant 弹簧劲度系数 $k$ measures stiffness -- a bigger $k$ is a harder spring.
- The displacement 位移 $x$ is measured from the spring's natural rest length.
- On a graph of force versus displacement, $k$ is simply the slope.
劲度与位移
- 弹簧劲度系数 $k$ 衡量弹簧的硬度——$k$ 越大,弹簧越硬。
- 位移 $x$ 是从弹簧的自然静止长度量起的。
- 在力对位移的图上,$k$ 就是斜率。
Hooke's law: F = -kx · 胡克定律:F = -kx
Increase the force and watch the spring stretch in proportion -- the slope is the spring constant k. · 增大力,看着弹簧成正比地伸长——斜率就是弹簧劲度系数 k。
A $6\ \text{N}$ force stretches a spring by $0.30\ \text{m}$. What is its spring constant (in N/m)? · 一个 $6\ \text{N}$ 的力把弹簧拉伸 $0.30\ \text{m}$。它的弹簧劲度系数是多少(单位 N/m)?
$k = F/x = 6/0.30 = 20\ \tfrac{\text{N}}{\text{m}}$. · $k = F/x = 6/0.30 = 20\ \tfrac{\text{N}}{\text{m}}$。
On a force-versus-displacement graph, the spring constant is the ____ of the line. · 在力对位移的图上,弹簧劲度系数是这条线的____。
Since $F = kx$ (magnitude), the graph is a straight line whose slope is $k$. · 由于 $F = kx$(大小),图是一条直线,其斜率为 $k$。
A restoring force
- Because it always points back to the middle, the spring force is a restoring force 回复力.
- Stretch right → it pulls left; compress left → it pushes right.
- That constant "return to center" is exactly what drives a mass on a spring to oscillate.
回复力
- 因为它总是指回中间,弹簧力是一种回复力。
- 向右拉 → 它向左拉;向左压 → 它向右推。
- 这种始终"回到中心"的性质,正是驱动弹簧上质量振动的原因。
You stretch a spring to the right. The spring force points... · 你把弹簧向右拉。弹簧力指向……
A restoring force always opposes the displacement -- stretch right, it pulls left. · 回复力总是反抗位移——向右拉,它向左拉。
The minus sign in $F_s = -kx$ shows that the spring force opposes the displacement. · $F_s = -kx$ 中的负号表明弹簧力反抗位移。
Yes -- the sign encodes the "restoring" direction, always back toward equilibrium. · 是的——这个符号编码了"回复"方向,总是指回平衡位置。
Combining springs
- Springs in parallel 并联 (side by side) act stiffer: $k_{eq} = k_1 + k_2$.
- Springs in series 串联 (end to end) act softer: $\dfrac{1}{k_{eq}} = \dfrac{1}{k_1} + \dfrac{1}{k_2}$.
- It is the exact opposite of how resistors combine.
弹簧的组合
- 并联的弹簧(并排)更硬:$k_{eq} = k_1 + k_2$。
- 串联的弹簧(首尾相接)更软:$\dfrac{1}{k_{eq}} = \dfrac{1}{k_1} + \dfrac{1}{k_2}$。
- 这与电阻的组合方式恰好相反。
Two identical springs of $k = 100\ \tfrac{\text{N}}{\text{m}}$ are placed in parallel. What is the equivalent spring constant (in N/m)? · 两根相同的 $k = 100\ \tfrac{\text{N}}{\text{m}}$ 弹簧并联。等效弹簧劲度系数是多少(单位 N/m)?
Parallel springs add, giving $k_{eq} = 100 + 100 = 200\ \tfrac{\text{N}}{\text{m}}$ -- stiffer together. · 并联弹簧相加,得 $k_{eq} = 100 + 100 = 200\ \tfrac{\text{N}}{\text{m}}$——合在一起更硬。
A spring stretches $0.20\ \text{m}$ when a $10\ \text{N}$ force pulls it.
- Spring constant: $k = \dfrac{F}{x} = \dfrac{10}{0.20} = 50\ \tfrac{\text{N}}{\text{m}}$.
- Stretch it twice as far ($0.40\ \text{m}$) and it pulls back with $20\ \text{N}$ -- always toward its rest position.
一个 $10\ \text{N}$ 的力拉弹簧,弹簧伸长 $0.20\ \text{m}$。
- 弹簧劲度系数:$k = \dfrac{F}{x} = \dfrac{10}{0.20} = 50\ \tfrac{\text{N}}{\text{m}}$。
- 拉远一倍($0.40\ \text{m}$),它就以 $20\ \text{N}$ 回拉——始终朝着它的静止位置。
The minus sign in $F_s = -kx$ is not decoration -- it encodes the direction. The spring force is never in the direction you pulled; it always opposes the displacement. Drop the sign and you lose the whole "restoring" idea.
$F_s = -kx$ 中的负号不是装饰——它编码了方向。弹簧力从不朝你拉的方向;它总是反抗位移。去掉这个符号,你就失去了整个"回复"的含义。
Hooke's law: $F_s = -kx$ -- a spring's force is proportional to its displacement and always points back to equilibrium (a restoring force). The spring constant $k$ is its stiffness (the slope of $F$ vs $x$). Parallel springs add stiffness; series springs soften.
胡克定律:$F_s = -kx$——弹簧的力与其位移成正比,并总是指回平衡位置(一种回复力)。弹簧劲度系数 $k$ 是它的硬度($F$-$x$ 图的斜率)。并联弹簧增加劲度;串联弹簧变软。