Forces and equilibrium · 力与平衡
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| force/fɔːs/ | 力 | lì |
| equilibrium/ˌiːkwɪˈlɪbrɪəm/ | 平衡 | píng héng |
| component/kəmˈpəʊnənt/ | 分量 | fèn liàng |
| friction/ˈfrɪkʃn/ | 摩擦力 | mó cā lì |
| normal reaction/ˈnɔːml rɪˈækʃn/ | 法向反作用力 | fǎ xiàng fǎn zuò yòng lì |
| limiting friction/ˈlɪmɪtɪŋ ˈfrɪkʃn/ | 最大静摩擦力 | zuì dà jìng mó cā lì |
| coefficient of friction/ˌkəʊɪˈfɪʃənt ɒv ˈfrɪkʃn/ | 摩擦系数 | mó cā xì shù |
| pulley/ˈpʊli/ | 滑轮 | huá lún |
| acceleration/əkˌseləˈreɪʃn/ | 加速度 | jiā sù dù |
| tension/ˈtenʃn/ | 张力 | zhāng lì |
Why bridges don't fall down
- Every bridge, every building, every shelf stays put because the forces 力 on it balance.
- Engineers call this equilibrium 平衡 — the vector sum of all forces is zero. Master this and you can analyse any static structure.
为什么桥不会倒
- 每座桥、每栋楼、每个架子都待在原地,因为作用在它上面的力平衡。
- 工程师把这叫做平衡(equilibrium)——所有力的向量和是零。掌握这个,你就能分析任何静态结构。
Adding forces · 力的相加
resultant = a + b
Forces add tip-to-tail. They are in equilibrium · 平衡 when the resultant is zero. · 力首尾相接地相加。当合力为零时它们处于平衡。
Forces and components 分量
- A force is a vector (size + direction). Split it into components: horizontal $F\cos\theta$, vertical $F\sin\theta$.
- A particle is in equilibrium when the forces balance — the vector sum is zero (components add to zero in every direction).
Worked example. A 10 N force acts at $30^{\circ}$ to the horizontal. Horizontal component $= 10\cos 30^{\circ} = 8.66$ N. Vertical component $= 10\sin 30^{\circ} = 5$ N.
A force splits into F cos theta across and F sin theta up
力与分量
- 一个力(force)是一个向量(大小 + 方向)。把它拆成分量(components):水平 $F\cos\theta$,垂直 $F\sin\theta$。
- 当力平衡时,一个质点处于平衡——向量和是零(在每个方向上分量加起来为零)。
算例。 一个 10 N 的力以与水平 $30^{\circ}$ 的角作用。水平分量 $= 10\cos 30^{\circ} = 8.66$ N。垂直分量 $= 10\sin 30^{\circ} = 5$ N。
一个力拆成横向的 F cos theta 和向上的 F sin theta
A particle is in equilibrium when the forces on it: · 一个质点处于平衡,当作用在它上的力:
Equilibrium means the forces balance — their vector sum is zero. · 平衡意味着力平衡——它们的向量和是零。
The horizontal component of a force F at angle θ to the horizontal is: · 一个与水平成角 θ 的力 F 的水平分量是:
Horizontal component = F cos θ; vertical component = F sin θ. · 水平分量 = F cos θ;垂直分量 = F sin θ。
Friction 摩擦力
- The contact force has the normal reaction 法向反作用力 $R$ (perpendicular) and friction $F$ (along the surface, opposing sliding).
- Friction grows only up to a maximum (limiting friction 最大静摩擦力):
- where $\mu$ is the coefficient of friction 摩擦系数.
Friction opposes motion (or tendency to move). If a block is about to slide down a slope, friction acts up the slope. The direction is always opposite to the (attempted) motion.
The contact force: normal reaction R up and friction F along the surface, with F at most mu R
摩擦力
- 接触力有法向反作用力(normal reaction)$R$(垂直)和摩擦力(friction)$F$(沿表面,阻碍滑动)。
- 摩擦力只增长到一个最大值(极限摩擦力,limiting friction):
- 其中 $\mu$ 是摩擦系数(coefficient of friction)。
摩擦力阻碍运动(或运动趋势)。 如果一个块即将沿斜面下滑,摩擦力沿斜面向上作用。方向总是与(企图的)运动相反。
接触力:向上的法向反作用力 R 和沿表面的摩擦力 F,F 最多为 mu R
The coefficient of friction is 0.4 and the normal reaction is 100 N. What is the maximum (limiting) friction force, in N? · 摩擦系数是 0.4,法向反作用力是 100 N。最大(极限)摩擦力是多少 N?
Maximum friction = μR = 0.4 × 100 = 40 N. · 最大摩擦力 = μR = 0.4 × 100 = 40 N。
Friction always acts in the direction of motion. · 摩擦力总是沿运动方向作用。
Friction opposes motion (or the tendency to move). It acts in the opposite direction to the (attempted) motion. · 摩擦力阻碍运动(或运动趋势)。它沿与(企图的)运动相反的方向作用。
Worked example — equilibrium on a slope
- A 5 kg block rests on a rough slope at $20^{\circ}$ to the horizontal. Find the friction force.
- Resolve parallel to slope: $F = mg\sin 20^{\circ} = 5 \times 10 \times 0.342 = 17.1$ N.
- Resolve perpendicular: $R = mg\cos 20^{\circ} = 5 \times 10 \times 0.940 = 47.0$ N.
On a slope, resolve the weight into components parallel (mg sin θ) and perpendicular (mg cos θ) to the surface.
算例——斜面上的平衡
- 一个 5 kg 的块静止在与水平 $20^{\circ}$ 的粗糙斜面上。求摩擦力。
- 沿斜面平行方向分解:$F = mg\sin 20^{\circ} = 5 \times 10 \times 0.342 = 17.1$ N。
- 垂直方向分解:$R = mg\cos 20^{\circ} = 5 \times 10 \times 0.940 = 47.0$ N。
在一个斜面上,把重力分解成平行(mg sin θ)和垂直(mg cos θ)于表面的分量。
A 5 kg block on a 30° slope. The component of weight parallel to the slope is mg sin 30°. Find it (g = 10). · 一个 5 kg 的块在一个 30° 的斜面上。重力平行于斜面的分量是 mg sin 30°。求它(g = 10)。
5 × 10 × sin 30° = 50 × 0.5 = 25 N. · 5 × 10 × sin 30° = 50 × 0.5 = 25 N。
A 5 kg block on a 30° slope. The normal reaction is mg cos 30°. Find it (g = 10, 1 dp). · 一个 5 kg 的块在一个 30° 的斜面上。法向反作用力是 mg cos 30°。求它(g = 10,1 位小数)。
5 × 10 × cos 30° = 50 × 0.866 = 43.3 N. · 5 × 10 × cos 30° = 50 × 0.866 = 43.3 N。
Connected particles
- When two particles are connected by a light string over a smooth pulley 滑轮, they have the same acceleration 加速度 and the same tension 张力.
- Write $F = ma$ for each particle, then solve the two equations simultaneously.
Forces on a block on a rough slope: weight, normal reaction, tension and friction
- Friction reaches its maximum in limiting equilibrium, when the body is about to slip.
连接的质点
- 当两个质点由一根轻绳经一个光滑滑轮连接时,它们有同样的加速度和同样的张力。
- 对每个质点写 $F = ma$,然后联立解这两个方程。
粗糙斜面上一个块受的力:重力、法向反作用力、张力和摩擦力
- 摩擦在极限平衡(limiting equilibrium)时达到最大,此时物体将要滑动。
You've got it
- a force splits into $F\cos\theta$ (horizontal) and $F\sin\theta$ (vertical)
- equilibrium = forces balance (vector sum zero)
- friction $F \leq \mu R$, reaching $\mu R$ when about to slip
你掌握了
- 一个力拆成 $F\cos\theta$(水平)和 $F\sin\theta$(垂直)
- 平衡 = 力平衡(向量和为零)
- 摩擦力 $F \leq \mu R$,在即将滑动时达到 $\mu R$