Newton's laws of motion · 牛顿运动定律
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| resultant force/rɪˈzʌltənt fɔːs/ | 合力 | hé lì |
| weight/weɪt/ | 重力 | zhòng lì |
| tension/ˈtenʃn/ | 张力 | zhāng lì |
| inclined plane/ɪnˈklaɪnd pleɪn/ | 斜面 | xié miàn |
| connected particles/kəˈnektɪd ˈpɑːtɪklz/ | 连接质点 | lián jiē zhì diǎn |
| inextensible/ˌɪnekˈstensɪbl/ | 不可伸长 | bù kě shēn cháng |
The law that launched rockets
- Every rocket, every car, every falling apple obeys the same rule: $F = ma$.
- Newton's second law is the foundation of all mechanics. Master it and you can predict the motion of anything.
发射火箭的定律
- 每枚火箭、每辆车、每个落下的苹果都服从同一条规则:$F = ma$。
- 牛顿第二定律是所有力学的基础。掌握它,你就能预测任何东西的运动。
Newton's second law
- The key law: resultant force 合力 = mass × acceleration:
- Weight 重力 is the force of gravity: $W = mg$ (take $g = 10\text{ m s}^{-2}$).
- Forces include weight, friction, tension 张力 in a string, and the push in a rod.
Worked example. A 4 kg box is pushed with a horizontal force of 20 N on a smooth surface. $F = ma \Rightarrow 20 = 4a \Rightarrow a = 5\text{ m s}^{-2}$.
Resultant force, not just any force. $F = ma$ uses the net (resultant) force. If there's friction, you must subtract it: $F_{\text{net}} = F_{\text{applied}} - F_{\text{friction}}$.
牛顿第二定律
- 关键的定律:合力 = 质量 × 加速度:
- 重力(weight)是引力:$W = mg$(取 $g = 10\text{ m s}^{-2}$)。
- 力包括重力、摩擦力、绳中的张力(tension),和杆中的推力。
算例。 一个 4 kg 的箱子在光滑表面上被一个 20 N 的水平力推。$F = ma \Rightarrow 20 = 4a \Rightarrow a = 5\text{ m s}^{-2}$。
是合力,不是任何一个力。 $F = ma$ 用净(合)力。如果有摩擦力,你必须减去它:$F_{\text{net}} = F_{\text{applied}} - F_{\text{friction}}$。
Resultant force · 合力
F = ma
The resultant force sets the acceleration. Balanced forces ⇒ no acceleration. · 合力决定加速度。平衡的力 ⇒ 没有加速度。
A resultant force gives a 5 kg mass an acceleration of 2 m/s². What is the force (F = ma), in N? · 一个合力给一个 5 kg 的质量 2 m/s² 的加速度。力是多少(F = ma),以 N 计?
F = ma = 5 × 2 = 10 N. · F = ma = 5 × 2 = 10 N。
Taking g = 10 m/s², what is the weight of a 2 kg object (W = mg), in N? · 取 g = 10 m/s²,一个 2 kg 物体的重力是多少(W = mg),以 N 计?
W = mg = 2 × 10 = 20 N. · W = mg = 2 × 10 = 20 N。
F = ma uses the resultant (net) force, not just any single force. · F = ma 用合(净)力,不是任何单一的力。
F = ma requires the net force. If friction is present, subtract it from the applied force first. · F = ma 需要净力。如果有摩擦力,先从施加的力中减去它。
Slopes and inclined planes 斜面
- On an inclined plane: split forces along and perpendicular to the slope, then apply $F = ma$ along the slope.
- Component of weight along slope: $mg\sin\theta$. Component perpendicular: $mg\cos\theta$.
斜坡与斜面
- 在一个斜面(inclined plane)上:把力沿斜面和垂直于斜面分解,然后沿斜面应用 $F = ma$。
- 重力沿斜面的分量:$mg\sin\theta$。垂直分量:$mg\cos\theta$。
To analyse motion on an inclined plane, you should: · 要分析斜面上的运动,你应该:
Resolve along the slope (apply F = ma there) and perpendicular to it (for the normal reaction). · 沿斜面分解(在那里应用 F = ma)和垂直于斜面分解(为法向反作用力)。
Connected particles 连接质点
- For connected particles (joined by a string): apply $F = ma$ to each body, or to the whole system.
- The string has the same tension throughout (if light and inextensible 不可伸长).
- Both bodies have the same acceleration.
Connected particles share one tension T and one acceleration; apply F = ma to each mass.
连接的质点
- 对连接的质点(由一根绳连接):对每个物体、或对整个系统应用 $F = ma$。
- 绳全程有同样的张力(如果轻且不可伸长)。
- 两个物体有同样的加速度。
连接的质点共享一个张力 T 和一个加速度;对每个质量应用 F = ma。
A 3 kg and 2 kg mass over a smooth pulley. Acceleration = g/5. With g = 10, find a (m/s²). · 一个 3 kg 和一个 2 kg 的质量经一个光滑滑轮。加速度 = g/5。g = 10,求 a(m/s²)。
a = g/5 = 10/5 = 2 m/s². · a = g/5 = 10/5 = 2 m/s²。
For the 3 kg and 2 kg masses over a pulley (a = 2 m/s²), find the tension T from T − 2g = 2a (g = 10). · 对经一个滑轮的 3 kg 和 2 kg 质量(a = 2 m/s²),从 T − 2g = 2a 求张力 T(g = 10)。
T − 2(10) = 2(2) → T = 20 + 4 = 24 N. · T − 2(10) = 2(2) → T = 20 + 4 = 24 N。
Worked example — connected particles
- A 3 kg and 2 kg mass are connected by a light string over a smooth pulley. Find the acceleration.
- For the 3 kg mass (moving down): $3g - T = 3a$.
- For the 2 kg mass (moving up): $T - 2g = 2a$.
- Adding: $3g - 2g = 5a \Rightarrow a = \dfrac{g}{5} = 2\text{ m s}^{-2}$.
- Newton's laws of motion give $F=ma$; forces may include thrust and air resistance.
算例——连接的质点
- 一个 3 kg 和一个 2 kg 的质量由一根轻绳经一个光滑滑轮连接。求加速度。
- 对 3 kg 质量(向下移动):$3g - T = 3a$。
- 对 2 kg 质量(向上移动):$T - 2g = 2a$。
- 相加:$3g - 2g = 5a \Rightarrow a = \dfrac{g}{5} = 2\text{ m s}^{-2}$。
- 牛顿运动定律(Newton's laws of motion)给出 $F=ma$;受力可包含推力(thrust)和空气阻力(air resistance)。
You've got it
- F = ma; weight $W = mg$
- on a slope, resolve forces along and perpendicular to the slope
- for connected particles, apply $F = ma$ to each body or the whole system
你掌握了
- F = ma;重力 $W = mg$
- 在一个斜坡上,把力沿斜面和垂直于斜面分解
- 对连接的质点,对每个物体或整个系统应用 $F = ma$