Newton's Second Law · 牛顿第二定律
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| mass/mæs/ | 质量 | zhì liàng |
The same shove — a feather flies, a fridge barely budges
- Give a feather a gentle push and it shoots off; give a fridge the same push and it hardly moves.
- Same force, wildly different results — the difference is mass 质量.
- Newton's second law ties them together: force, mass and acceleration in one equation.
- It is the engine of mechanics: give me the forces and the mass, and I'll give you the motion.
同样一推——羽毛飞出去,冰箱几乎不动
- 轻推一根羽毛,它飞了出去;用同样的力推冰箱,它几乎不动。
- 同样的力,结果天差地别——差别在于质量。
- 牛顿第二定律把它们联系起来:力、质量和加速度写进一个方程。
- 它是力学的引擎:给我力和质量,我就给你运动。
The equation
- The net force equals mass times acceleration: $\vec F_{\text{net}} = m\,\vec a$.
- Rearranged: $\vec a = \dfrac{\vec F_{\text{net}}}{m}$ — acceleration is what the net force produces.
- The acceleration points in the same direction as the net force.
- It is the net force that counts, never one force in isolation.
方程
- 合力等于质量乘以加速度:$\vec F_{\text{net}} = m\,\vec a$。
- 变形:$\vec a = \dfrac{\vec F_{\text{net}}}{m}$——加速度就是合力所产生的。
- 加速度与合力方向相同。
- 起作用的是合力,绝不是孤立的某一个力。
What net force gives a $2\ \text{kg}$ object an acceleration of $5\ \tfrac{\text{m}}{\text{s}^2}$? Answer in $\text{N}$. · 什么合力能使一个$2\ \text{kg}$kg的物体产生$5\ \tfrac{\text{m}}{\text{s}^2}$m/s²的加速度？答案单位是$\text{N}$N。
$F = ma = 2 \times 5 = 10\ \text{N}$.
Two proportions inside one law
- More force, more acceleration: $a \propto F$ (double the force, double the acceleration).
- More mass, less acceleration: $a \propto \dfrac{1}{m}$ (double the mass, halve the acceleration).
- So a light cart leaps ahead while a heavy one crawls under the same push.
- Heavy objects are "sluggish" precisely because $a = F/m$ shrinks as $m$ grows.
一条定律里藏着两个正比
- 力越大,加速度越大:$a \propto F$(力加倍,加速度加倍)。
- 质量越大,加速度越小:$a \propto \dfrac{1}{m}$(质量加倍,加速度减半)。
- 所以同样一推下,轻车一跃而出,重车却慢慢爬。
- 重物"迟钝"正是因为 $a = F/m$ 随 $m$ 增大而变小。
Net force sets the acceleration · 合力决定加速度
Raise the applied force or the mass and watch how the net force and the acceleration change. · 增大施加的力或质量，观察合力和加速度的变化。
A net force of $12\ \text{N}$ acts on a $3\ \text{kg}$ trolley. What is its acceleration, in $\tfrac{\text{m}}{\text{s}^2}$? · 一个$12\ \text{N}$N的合力作用在一个$3\ \text{kg}$kg的小车上。它的加速度是多少，单位是$\tfrac{\text{m}}{\text{s}^2}$m/s²？
$a = F_{\text{net}}/m = 12/3 = 4\ \tfrac{\text{m}}{\text{s}^2}$.
If you double the net force on an object but keep its mass the same, its acceleration: · 如果你将作用在物体上的合力加倍但保持其质量不变，它的加速度会：
Since $a = F/m$ and $m$ is fixed, $a \propto F$: doubling the force doubles the acceleration. · 因为$a = F/m$F 和 $m$m 固定，根据$a \propto F$a=F/m：加倍力会使加速度加倍。
Doubling the mass while keeping the same net force halves the acceleration. · 在保持相同合力的情况下加倍质量，加速度会减半。
$a = F/m$, so $a \propto 1/m$: double the mass, half the acceleration. · $a = F/m$F=ma，所以$a \propto 1/m$a=F/m：质量加倍，加速度减半。
Units and weight
- $1\ \text{N} = 1\ \text{kg} \cdot \tfrac{\text{m}}{\text{s}^2}$ — the newton is defined by this law.
- Weight is just the second law applied to gravity: $W = mg$.
- Mass ($\text{kg}$) measures inertia; weight ($\text{N}$) is a force — do not confuse them.
- On the Moon your mass is unchanged, but your weight is about one-sixth.
单位与重力
- $1\ \text{N} = 1\ \text{kg} \cdot \tfrac{\text{m}}{\text{s}^2}$——牛顿正是由这条定律定义的。
- 重力就是把第二定律用于引力:$W = mg$。
- 质量($\text{kg}$)量度惯性;重力($\text{N}$)是一个力——不要混淆。
- 在月球上你的质量不变,但你的重力大约只有六分之一。
Complete the definition: $1\ \text{N} = 1\ \text{kg} \cdot$ ____ (write the units). · 完成定义：$1\ \text{N} = 1\ \text{kg} \cdot$ ____（写出单位）。
From $F = ma$, $1\ \text{N} = 1\ \text{kg} \cdot 1\ \tfrac{\text{m}}{\text{s}^2}$. · 从$F = ma$F=ma可知，$1\ \text{N} = 1\ \text{kg} \cdot 1\ \tfrac{\text{m}}{\text{s}^2}$a=F/m。
You travel from Earth to the Moon. Which of these changes? · 你从地球前往月球。以下哪项发生了变化？
Mass (inertia) is unchanged; weight $W = mg$ drops because the Moon's $g$ is about one-sixth of Earth's. · 质量（惯性）保持不变；重量$W = mg$减小，因为月球的$g$重力加速度约为地球的六分之一。
Use the net force in $F = ma$, not a single force. If several forces act, add them as vectors first. And remember mass is not weight: mass is in kilograms, weight is a force in newtons ($W = mg$).
在 $F = ma$ 中要用合力,不是某一个力。若有几个力作用,先把它们按矢量相加。并且记住质量不是重力:质量的单位是千克,重力是以牛顿为单位的力($W = mg$)。
A net force of $12\ \text{N}$ acts on a $3\ \text{kg}$ trolley.
- $a = \dfrac{F_{\text{net}}}{m} = \dfrac{12}{3} = 4\ \tfrac{\text{m}}{\text{s}^2}$.
Keep the same force but double the mass to $6\ \text{kg}$, and the acceleration halves to $2\ \tfrac{\text{m}}{\text{s}^2}$.
一个 $12\ \text{N}$ 的合力作用在一辆 $3\ \text{kg}$ 的小车上。
- $a = \dfrac{F_{\text{net}}}{m} = \dfrac{12}{3} = 4\ \tfrac{\text{m}}{\text{s}^2}$。
保持力不变,把质量加倍到 $6\ \text{kg}$,加速度就减半到 $2\ \tfrac{\text{m}}{\text{s}^2}$。
Newton's second law: $\vec F_{\text{net}} = m\vec a$, so $\vec a = \vec F_{\text{net}}/m$ in the direction of the net force. Acceleration grows with force ($a\propto F$) and shrinks with mass ($a\propto 1/m$). $1\ \text{N} = 1\ \text{kg}\cdot\text{m/s}^2$; weight is $W = mg$.
牛顿第二定律:$\vec F_{\text{net}} = m\vec a$,所以 $\vec a = \vec F_{\text{net}}/m$,方向沿合力。加速度随力增大($a\propto F$)、随质量减小($a\propto 1/m$)。$1\ \text{N} = 1\ \text{kg}\cdot\text{m/s}^2$;重力是 $W = mg$。