Newton's laws, force and momentum · 牛顿定律、力与动量
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| momentum/məʊˈmentəm/ | 动量 | dòngliàng |
| impulse/ˈɪmpʌls/ | 冲量 | chōng liàng |
| collision/kəˈlɪʒn/ | 碰撞 | pèng zhuàng |
| weight/weɪt/ | 重力 | zhònglì |
| resultant force/rɪˈzʌltənt fɔːs/ | 合力 | hé lì |
| inertia/ɪˈnɜːʃə/ | 惯性 | guàn xìng |
Why soft hands catch an egg
- Catch a thrown egg with stiff hands and it breaks; with soft, giving hands it survives.
- Both stop the same egg — but soft hands take more time, so the force is smaller.
- That link between force, momentum 动量 and time is the heart of dynamics.
为什么柔软的手能接住鸡蛋
- 用僵硬的手接一个抛来的鸡蛋,它会破;用柔软、能缓冲的手接,它就没事。
- 两者都让同一个鸡蛋停下——但柔软的手用了 更长的时间,所以 力 更小。
- 力、动量和时间之间的这种联系,是动力学的核心。
Momentum
- Momentum $p = mv$ — mass times velocity. It is a vector (same direction as $v$).
- Unit: $\dfrac{\text{kg}\cdot\text{m}}{\text{s}}$, the same as $\text{N}\cdot\text{s}$.
A rocket pushes gas down; by Newton's third law the gas pushes it up
动量
- 动量(momentum) $p = mv$——质量乘以速度。它是 矢量(方向与 $v$ 相同)。
- 单位:$\dfrac{\text{kg}\cdot\text{m}}{\text{s}}$,与 $\text{N}\cdot\text{s}$ 相同。
火箭把气体向下喷;根据牛顿第三定律,气体把火箭向上推
Free-body diagram (F = ma) · 受力图(F = ma)
The resultant of the forces, divided by the mass, gives the acceleration. · 合力除以质量,得到加速度。
Newton's second law · 牛顿第二定律
F = ma (resultant) · F = ma(合力)
The resultant force sets the acceleration; balanced forces ⇒ none. · 合力 决定加速度;力平衡 ⇒ 没有加速度。
A $2.0\ \text{kg}$ ball moves at $3.0\ \dfrac{\text{m}}{\text{s}}$. What is its momentum? · 一个 $2.0\ \text{kg}$ 的球以 $3.0\ \dfrac{\text{m}}{\text{s}}$ 运动。它的动量是多少?
$p = mv = 2.0 \times 3.0 = 6.0\ \dfrac{\text{kg}\cdot\text{m}}{\text{s}}$. · $p = mv = 2.0 \times 3.0 = 6.0\ \dfrac{\text{kg}\cdot\text{m}}{\text{s}}$。
Momentum is a vector. · 动量是矢量。
Yes — $p = mv$ points the same way as the velocity, so it has a direction. · 是的——$p = mv$ 与速度方向相同,所以它有方向。
Force = rate of change of momentum
- Newton's second law, in full: $F = \dfrac{\Delta p}{\Delta t}$.
- When the mass is constant this becomes the familiar $F = ma$.
In a crash, a large force acts over a very short time to change momentum
力 = 动量的变化率
- 牛顿第二定律的完整形式:$F = \dfrac{\Delta p}{\Delta t}$。
- 当质量恒定时,它就变成熟悉的 $F = ma$。
在碰撞中,一个很大的力在极短的时间内作用,改变动量
In its general form, Newton's second law says the resultant force equals the: · 在一般形式下,牛顿第二定律说合力等于:
$F = \dfrac{\Delta p}{\Delta t}$ — the rate of change of momentum. For constant mass this is $F = ma$. · $F = \dfrac{\Delta p}{\Delta t}$——动量的变化率。质量恒定时这就是 $F = ma$。
Match each of Newton's laws to its statement. · 把牛顿的每条定律与它的表述配对。
The second law's wording is about the rate of change of MOMENTUM. F = ma is a special case, valid only when the mass is constant. · 第二定律的措辞讲的是动量的变化率。F = ma 是特例,只在质量恒定时成立。
Impulse 冲量 and collisions 碰撞
- The average force in a hit is $F = \dfrac{\Delta p}{\Delta t}$ — change in momentum over contact time.
- A longer contact time means a smaller force (the egg, a crumple zone, a bent knee on landing).
- Energy link: $E_{\text{k}} = \dfrac{p^{2}}{2m}$.
Weight 重力 and normal contact force on a book resting on a table
冲量与碰撞
- 一次撞击中的平均力是 $F = \dfrac{\Delta p}{\Delta t}$——动量变化除以接触时间。
- 更长 的接触时间意味着 更小 的力(鸡蛋、汽车的溃缩区、落地时弯曲的膝盖)。
- 能量关系:$E_{\text{k}} = \dfrac{p^{2}}{2m}$。
放在桌上的书所受的重力和法向接触力
Catching an egg with soft hands makes the contact time longer, which makes the average force ____. · 用柔软的手接鸡蛋会让接触时间变长,从而使平均力 ____。
Same change in momentum over a longer time means a smaller force: $F = \dfrac{\Delta p}{\Delta t}$. · 相同的动量变化在更长的时间内意味着更小的力:$F = \dfrac{\Delta p}{\Delta t}$。
A 0.15 kg ball hits a wall at 12 m/s and rebounds at 8.0 m/s along the same line. What is the magnitude of the impulse on it, in N s? · 一个 0.15 kg 的球以 12 m/s 撞墙,沿同一直线以 8.0 m/s 反弹。它受到的冲量大小是多少 N s?
Take one direction as positive: the change is 0.15 x (8.0 - (-12)) = 3.0 N s. Adding the speeds without the sign gives the same number here only because the ball reverses; get the sign convention right or a glancing rebound will catch you out. · 取一个方向为正:变化量是 0.15 x (8.0 - (-12)) = 3.0 N s。这里不管符号直接把速率相加也得同一个数,只因为球是原路反弹;符号约定要做对,否则斜碰的题就会出错。
Newton's first law
- With zero resultant force 合力, an object stays at rest or moves at constant velocity.
- You lurch forward when a bus brakes because your body "wants" to keep moving — inertia 惯性.
牛顿第一定律
- 当 合力为零 时,物体保持静止或匀速运动。
- 公交车刹车时你会前倾,因为你的身体“想”继续运动——惯性(inertia)。
Newton's second law
- A resultant force gives an acceleration: $F = ma$.
- The acceleration always points the same way as the resultant force.
牛顿第二定律
- 合力产生加速度:$F = ma$。
- 加速度总是指向与合力 相同的方向。
Newton's third law
- If A pushes B, then B pushes A back — equal in size, opposite in direction.
- The two forces act on different objects and are the same type of force.
牛顿第三定律
- 如果 A 推 B,那么 B 也推 A——大小相等、方向相反。
- 这两个力作用在 不同的物体 上,且是 同一类型 的力。
A book's weight and the table's upward push are not a third-law pair — they act on the same object (the book) and are different types of force.
书的 重力 和桌子的 向上的推力 不是 一对第三定律的力——它们作用在 同一个 物体(书)上,而且是不同类型的力。
Which pair are genuine Newton's third-law partners? · 哪一对是真正的牛顿第三定律的力对?
The contact forces "book on table" and "table on book" are equal, opposite, same type, and act on different objects. The book's weight and the table's push act on the same · 相同 object, so they are not a pair. · 接触力“书对桌”和“桌对书”大小相等、方向相反、同类型,且作用在不同物体上。书的重力和桌子的推力作用在同一个物体上,所以它们不是一对。
Select all · 所有 the true statements about a Newton's third-law pair of forces. · 选出所有关于牛顿第三定律力对的正确说法。
A third-law pair is equal, opposite, the same type, and acts on two different objects — never on the same one. · 第三定律力对大小相等、方向相反、同类型,并作用在 两个不同的 物体上——绝不会作用在同一个上。
Weight
- Weight is the gravitational force on a mass: $W = mg$.
- Mass is the same everywhere; weight changes with $g$ (less on the Moon).
重力(重量)
- 重量(weight) 是作用在质量上的引力:$W = mg$。
- 质量处处相同;重量随 $g$ 变化(在月球上更小)。
What is the weight of a $5.0\ \text{kg}$ bag near Earth's surface? (Use $g = 9.81\ \dfrac{\text{m}}{\text{s}^2}$.) · 在地球表面附近,一个 $5.0\ \text{kg}$ 的袋子的重力是多少?(取 $g = 9.81\ \dfrac{\text{m}}{\text{s}^2}$。)
$W = mg = 5.0 \times 9.81 \approx 49\ \text{N}$. · $W = mg = 5.0 \times 9.81 \approx 49\ \text{N}$。
For a body in equilibrium, do not say the forces cancel. Say the ____ force on it is zero. · 物体处于平衡时,不要说力互相抵消。要说它所受的____为零。
"Balanced" and "cancel out" are both refused. The examiner is checking that you can name the single vector sum of all the forces. · "平衡"和"抵消"都不被接受。考官要看的是你能否说出所有力的那个矢量和。
Put the test for a Newton's third-law pair in order. · 把判断牛顿第三定律一对力的检验按顺序排列。
The last two are what rule out the classic wrong answer. A book's weight (gravitational) and the table's push on it (contact) fail both: same body, different types. · 最后两条正是用来排除那个经典错误答案的。书的重力(引力)与桌子对书的支持力(接触力)两条都不满足:同一个物体,不同的类型。
Marks that slip away
- Never write "the forces are balanced" or "the forces cancel out". The marked phrase is the resultant force on the object is zero.
- Newton's second law is about the rate of change of momentum, not simply $F = ma$. That form assumes a constant mass.
- A third-law pair acts on two different objects and is one type of force. The weight of a book and the table's push on it are not a pair: both act on the book.
- Conservation of momentum needs its condition stated: provided no resultant external force acts on the system.
- Mass resists a change in motion and is in kilograms; weight is a force in newtons, equal to $mg$. A question that says "mass" wants kilograms.
容易丢掉的分
- 绝不要写"力是平衡的"或"力互相抵消"。评分认可的说法是物体所受的合力为零。
- 牛顿第二定律讲的是动量的变化率,不只是 $F = ma$。那个形式假定质量恒定。
- 第三定律的一对力作用在两个不同的物体上,而且是同一种类型的力。书的重力和桌子对书的支持力不是一对:两者都作用在书上。
- 动量守恒必须把条件说出来:在系统不受合外力的前提下。
- 质量是抵抗运动状态改变的属性,单位是千克;重量是力,单位是牛顿,等于 $mg$。题目说"质量"就要用千克。
You've got it
- momentum $p = mv$ (a vector); force $F = \dfrac{\Delta p}{\Delta t} = ma$
- a longer contact time gives a smaller force (impulse)
- third-law pairs are equal, opposite, same type, different objects; $W = mg$
你掌握了
- 动量 $p = mv$(矢量);力 $F = \dfrac{\Delta p}{\Delta t} = ma$
- 更长 的接触时间给出 更小 的力(冲量)
- 第三定律的力对:大小相等、方向相反、同类型、不同物体;$W = mg$