Mechanics
A-Level Mathematics Topic 4 12:38 English narration · English + 中文 subtitles burned in
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Push a shopping trolley, and it speeds up.
推一辆购物车，它会加速。
Push harder, and it speeds up faster.
推得更用力，它加速得更快。
Push a heavier one with the same effort, and it barely moves.
用同样的力气去推一辆更重的， 它几乎不动。
All of mechanics grows from that one everyday truth, captured in three symbols: force equals mass times acceleration.
整个力学，都从这个日常的真理生长出来，浓缩在三个符号里： 力等于质量乘以加速度。
Double the force, and you double the acceleration.
力加倍，加速度也加倍。
Master this single law, and you can predict how anything moves.
掌握这一条定律，你就能预测任何东西如何运动。
Mechanics is the mathematics of the physical world — of forces, of motion, and of energy.
力学是关于物理世界的数学——关于力、关于运动、关于能量。
We will learn to draw the forces on an object, to track how it moves, to follow momentum through a collision, and to trace energy as it changes form.
我们将学会画出物体上的受力， 追踪它如何运动，跟随动量穿过一次碰撞，并追迹能量如何改变形式。
Every problem starts the same way: with a clear picture.
每一道题都以同样的方式开始：从一张清晰的图开始。
Let's begin.
让我们开始吧。
Throughout this topic, take the acceleration of free fall as ten metres per second squared.
在本专题中，始终取自由落体加速度为每秒平方十米。
That keeps every number clean.
这样每个数字都干净利落。
A force is a vector — it has size and direction — so several forces on one body can be added into one resultant force.
力是向量—— 既有大小也有方向——所以作用在同一物体上的几个力，可以合成一个合力。
That single vector replaces them all, and it is what you use in the laws of motion.
那一个向量就代替了全部的力，也是你在运动定律里要用的量。
A force is a push or a pull, and like every vector it has both a size and a direction.
力是一种推或拉，和每一个向量一样，它同时具有大小和方向。
That means we can split any slanted force into two components — one horizontal, one vertical — using the cosine and the sine of its angle.
这意味着我们可以把任何倾斜的力， 拆成两个分量——一个水平，一个竖直——用它角度的余弦和正弦来算。
This is the key move in almost every problem.
这是几乎每道题的关键一步。
And when an object sits perfectly still, its forces are in equilibrium: they cancel out.
而当一个物体完全静止时，它受的力处于平衡：彼此抵消。
The components in every direction must add to zero.
每个方向上的分量之和，都必须为零。
Look at the right triangle.
看这个直角三角形。
The force is the hypotenuse.
力是斜边。
The horizontal part is F times cosine of theta; the vertical part is F times sine of theta.
水平部分是力乘以角度的余弦；竖直部分是力乘以角度的正弦。
Always resolve into a pair of perpendicular directions you choose — often horizontal and vertical, or along a slope and across it.
永远分解成你选定的一对垂直方向——常常是水平和竖直，或沿斜面与垂直于斜面。
Once the force is split, each direction becomes a simple one-dimensional sum.
力一旦拆开，每个方向就变成简单的一维求和。
Every mechanics problem is a picture like this.
每一道力学题，都是这样一张图。
Three forces act on the climber: weight straight down, tension along the rope, and a push from the rock.
攀岩者身上有三个力：竖直向下的重力、沿绳的张力， 以及岩壁的推力。
Because they balance, the climber hangs still in equilibrium.
因为它们平衡，攀岩者静止悬挂。
Resolve each force into horizontal and vertical parts, set the sum in each direction to zero, and the picture becomes a pair of equations you can solve.
把每个力分解成水平和竖直分量， 令每个方向之和为零，这张图就变成你可以求解的两个方程。
When two surfaces touch, they press on each other with a contact force.
当两个表面接触时，它们以一个接触力相互挤压。
We split it into two parts: the normal reaction, pushing straight out of the surface, and friction, acting along it.
我们把它拆成两部分： 垂直于表面向外推的法向反作用力，以及沿着表面作用的摩擦力。
Friction always opposes sliding, but it can only grow so far.
摩擦力总是阻碍滑动， 但它只能增大到一定程度。
It rises up to a maximum of mu times the normal reaction.
它最多增大到摩擦系数乘以法向反作用力。
At that limit the object is on the very point of slipping.
在这个极限上，物体正处于将要滑动的临界点。
On a rough surface the contact force splits into the normal reaction R, straight up from the table, and friction F along it, at most mu R.
在粗糙表面上，接触力拆成竖直向上的法向反作用力，以及沿表面的摩擦力，最多是摩擦系数乘以法向力。
A smooth surface is a model with no friction at all — useful when the problem says so.
光滑表面是一个完全没有摩擦的模型——当题目这样说时很有用。
When friction reaches its maximum, the body is in limiting equilibrium, about to slip, and we write F equals mu R.
当摩擦达到最大值时， 物体处于极限平衡，即将滑动，我们写成摩擦力等于摩擦系数乘以法向力。
By Newton's third law, the two surfaces push on each other with equal and opposite forces.
根据牛顿第三定律，两个表面以大小相等、方向相反的力相互推挤。
Body A pushes on body B; body B pushes back on A with the same size of force, opposite in direction.
物体甲推物体乙； 物体乙以同样大小、相反方向的力推回甲。
The pair always comes together — never one force without its partner.
这对力总是成对出现——从不会只有一个力而没有它的伙伴。
Worked example.
例题。
A block of weight twenty newtons rests on a rough horizontal table with coefficient of friction zero point four.
一个重二十牛顿的物块放在粗糙水平桌面上，摩擦系数为零点四。
The table's normal reaction balances the weight, so R is twenty newtons.
桌面的法向反作用力 平衡重力，所以法向力是二十牛顿。
The block is on the point of slipping when friction reaches its maximum: mu times R is zero point four times twenty, which is eight newtons.
物块在摩擦力达到最大值时处于将滑临界： 摩擦系数乘以法向力是零点四乘二十，等于八牛顿。
In equilibrium the applied force equals that friction, so the largest horizontal force before the block slides is eight newtons.
平衡时施加的力等于这个摩擦力， 所以在物块滑动之前，最大水平力是八牛顿。
Before graphs, fix the language.
在画图之前，先把用语固定下来。
Distance and speed are scalars — they have size only.
距离和速率是标量——只有大小。
Displacement, velocity and acceleration are vectors — they have size and direction.
位移、速度和加速度是向量—— 既有大小也有方向。
Saying a car moves at twenty metres per second is speed; saying it moves twenty metres per second due east is velocity.
说一辆车以每秒二十米运动，那是速率；说它向东每秒二十米，那是速度。
Direction changes everything.
方向会改变一切。
To describe motion, the velocity-time graph is your best friend.
要描述运动，速度时间图是你最好的朋友。
Plot velocity against time, and it tells you everything.
把速度对时间画出来，它就把一切都告诉你。
The area underneath the graph is the distance travelled.
图线下方的面积，就是走过的距离。
And the gradient — the steepness — is the acceleration.
而图线的斜率——它的陡峭程度——就是加速度。
More generally, differentiate to step from displacement, to velocity, to acceleration; and integrate to step back the other way.
更一般地说，求导可以从位移，走到速度，再走到加速度；而积分则沿相反方向走回去。
The shaded area gives the distance travelled; the slope of each segment gives the acceleration.
阴影面积给出走过的距离；每一段的斜率给出加速度。
A rising line means speeding up; a flat line means constant speed; a falling line means slowing down — a negative acceleration, often called a deceleration.
上升的线表示加速；平线表示匀速； 下降的线表示减速——负的加速度，常称为减速度。
On a displacement-time graph, by contrast, the gradient is the velocity itself.
相比之下，在位移时间图上，斜率就是速度本身。
When the acceleration is constant, four famous formulas — the suvat equations — connect displacement, the initial and final velocity, the acceleration, and the time.
当加速度恒定时，四个著名的公式——匀加速运动方程——把位移、初速度和末速度、加速度和时间联系起来。
Pick the one that holds the three things you know, and the one you want.
挑出那个正好含有你已知的三个量、以及你要求的量的公式。
For instance, a car starting from rest, accelerating at two point five, for four seconds, reaches ten metres per second, and travels twenty metres.
例如，一辆车从静止出发， 以二点五的加速度，经过四秒，达到每秒十米，并驶过二十米。
How do you pick?
怎样挑选？
First list the three things you know.
第一，列出已知的三个量。
Second, name the one you want.
第二，写出你要求的量。
Third, choose the formula that holds exactly those four letters — each suvat equation leaves one letter out.
第三，选正好含有这四个字母的公式—— 每个匀加速方程都会漏掉一个字母。
Fourth, keep one consistent positive direction for the whole solution; if right is positive, left is negative.
第四，整道解保持一个一致的正方向；如果向右为正，向左就为负。
Never use suvat when the acceleration is changing.
当加速度在变化时，绝不要用这些方程。
A Newton's cradle is the classic picture of momentum in collisions.
牛顿摆是碰撞中动量的经典画面。
Pull one ball back, let it swing, and the ball on the far end flies out — while the middle ones barely move.
把一个球拉回再放开，远端的球飞出去——中间的几乎不动。
Linear momentum is mass times velocity.
线动量是质量乘以速度。
It is a vector, and in a direct collision of two bodies the total momentum is unchanged.
它是向量，在两个物体的正碰中，总动量不变。
Momentum is mass times velocity, and it too is a vector.
动量是质量乘以速度，它也是一个向量。
Its magic appears in collisions.
它的魔力出现在碰撞中。
Picture two bodies before they meet, each carrying its own momentum.
设想两个物体在相遇之前， 各自带着自己的动量。
They crash — perhaps sticking together — and move off after.
它们相撞——也许粘在一起——然后一起离开。
The grand rule is this: the total momentum before the collision equals the total momentum after.
宏大的法则是这样的： 碰撞前的总动量，等于碰撞后的总动量。
Momentum is conserved, always.
动量守恒，永远如此。
Write it as m one u one plus m two u two equals m one v one plus m two v two.
写成质量一乘初速一，加质量二乘初速二，等于质量一乘末速一，加质量二乘末速二。
The total momentum is the same before and after the collision.
碰撞前后总动量相同。
If the bodies stick together, they share one common velocity after impact — an inelastic collision that is still governed by momentum conservation.
如果两物体粘在一起，碰撞后它们共享同一个共同速度—— 这是非弹性碰撞，但仍然服从动量守恒。
Worked example.
例题。
A body of mass two kilograms moving at three metres per second hits a stationary body of mass one kilogram, and they stick together.
一个质量两千克、以每秒三米运动的物体，撞上一个静止的质量一千克的物体，并粘在一起。
Before the crash the total momentum is two times three, plus one times zero, which is six.
碰撞前总动量是二乘三，加一乘零，等于六。
After, the combined mass three times v equals that six.
碰撞后，合质量三乘以共同速度等于六。
So the common speed is two metres per second.
所以共同速度是每秒两米。
Newton's second law needs a clear list of forces.
牛顿第二定律需要一份清晰的力清单。
Weight is mass times gravity — ten times the mass in this topic.
重力是质量乘以重力加速度——本专题里是质量的十倍。
Friction opposes sliding, at most mu R.
摩擦力阻碍滑动，最多是摩擦系数乘以法向力。
A string pulls with tension along its length; a rod can push with thrust.
绳子沿其方向以张力拉；杆可以以推力推。
And air resistance, or drag, opposes motion through the air.
而空气阻力则阻碍在空气中的运动。
Name every force on your diagram before you write an equation.
在写方程之前，先在图上标出每一个力。
Now the heart of it all — Newton's second law.
现在是全部的核心——牛顿第二定律。
The resultant force on a body equals its mass times its acceleration.
物体所受的合力，等于它的质量乘以它的加速度。
Its weight is simply its mass times gravity.
它的重力，不过就是它的质量乘以重力加速度。
The classic challenge is an object on a slope.
经典的难题是一个放在斜面上的物体。
There, you resolve every force into a part along the slope, and a part at right angles to it.
在那里，你把每一个力都分解成沿斜面的一部分，和垂直于斜面的一部分。
Then apply force equals mass times acceleration, along the direction of motion.
然后沿运动方向，应用力等于质量乘以加速度。
On an inclined plane, resolve the forces along the slope and at right angles to it.
在斜面（inclined plane）上，把力沿斜面和垂直于斜面分解。
The weight splits into mg sine theta down the slope and mg cosine theta into the plane.
重力拆成沿斜面向下的质量乘重力加速度乘正弦， 以及压向斜面的质量乘重力加速度乘余弦。
The normal reaction balances the perpendicular part; friction and tension act along the slope.
法向反作用力平衡垂直部分；摩擦与张力沿斜面作用。
Write force equals mass times acceleration along the direction of motion.
沿运动方向写力等于质量乘以加速度。
Worked example.
例题。
A block of mass twelve kilograms is pulled up a rough plane by a rope parallel to the slope.
一个质量十二千克的物块，被平行于斜面的绳子沿粗糙斜面向上拉。
The plane is at twenty degrees to the horizontal, the coefficient of friction is zero point four, and the acceleration is two metres per second squared.
斜面与水平成二十度， 摩擦系数为零点四，加速度为每秒平方两米。
The normal reaction is one hundred and twenty times cosine of twenty, which is one hundred twelve point eight newtons, so friction is zero point four times that — forty-five point one newtons.
法向反作用力是一百二十乘以二十度的余弦， 即一百一十二点八牛顿，所以摩擦力是零点四乘它——四十五点一牛顿。
Along the slope, tension minus the component of weight minus friction equals mass times acceleration.
沿斜面， 张力减去重力沿斜面分量再减去摩擦力，等于质量乘加速度。
So tension is twenty-four plus forty-one plus forty-five point one, which is one hundred and ten newtons to three significant figures.
所以张力是二十四加四十一加四十五点一， 精确到三位有效数字是一百一十牛顿。
For connected particles — bodies joined by a string — apply force equals mass times acceleration to each body, or to the whole system.
对于连接质点——被绳子连在一起的物体——对每个物体，或对整体，应用力等于质量乘加速度。
If the string is light, the tension is the same at both ends.
如果绳子是轻的，两端张力相同。
If it is inextensible, both bodies share the same magnitude of acceleration.
如果绳子不可伸长，两个物体共享同样大小的加速度。
Two equations, one shared tension, one shared acceleration — that is the pattern.
两个方程，一个共享的张力，一个共享的加速度——这就是模式。
A roller coaster trades potential energy for kinetic energy as it rises and falls.
过山车在上升和下降时，在势能与动能之间交换。
At the top of a hill the coaster is high and slow — lots of potential, little kinetic.
在山顶它高而慢——势能多，动能少。
At the bottom it is low and fast — the potential has become motion.
在谷底它低而快——势能变成了运动。
When no friction acts, the total energy is conserved: it merely swaps between the two stores.
当没有摩擦时，总能量守恒：只是在两个储库之间交换。
Finally, energy.
最后，能量。
Work is done when a force moves something — force times the distance, measured in joules.
当一个力使某物移动时，就做了功——力乘以距离，以焦耳来量。
That work becomes energy, which comes in two great stores: kinetic energy, the energy of movement, and potential energy, the energy of height.
这份功变成能量，而能量有两大储库：动能，运动的能量；以及势能，高度的能量。
When no friction acts, the total energy is conserved — merely swapping between the two, as on a roller coaster.
当没有摩擦时，总能量守恒——只是在两者之间来回交换，就像过山车一样。
And power is how fast you do the work: force times velocity, measured in watts.
而功率，是你做功有多快：力乘以速度，以瓦特来量。
More carefully: the work done by a constant force is the scalar product of force and displacement — the force times the distance moved in the direction of the force, F d cosine theta, where theta is the angle between the force and the motion.
更仔细地说：恒力做的功，是力与位移的标量积（数量积）——力乘以沿力方向移动的距离， 也就是力乘距离乘以夹角的余弦，其中夹角是力与运动方向之间的角。
Kinetic energy is half m v squared.
动能是二分之一质量乘速度的平方。
Gravitational potential energy is m g h.
重力势能是质量乘重力加速度乘高度。
The work done by the outside forces equals the change in the total energy.
外力做的功，等于总能量的改变。
Worked example.
例题。
A car engine works at twelve kilowatts while the car moves at twenty metres per second on a level road.
一辆汽车的发动机以十二千瓦工作，同时汽车在水平路上以每秒二十米行驶。
Power equals force times velocity, so the driving force is power over velocity: twelve thousand divided by twenty.
功率等于力乘以速度，所以驱动力是功率除以速度：一万二千除以二十。
That is six hundred newtons.
那就是六百牛顿。
Before you go, four ways to keep your marks.
结束之前，四个保住分数的办法。
First, always draw a clear force diagram and resolve into perpendicular components; for equilibrium, each direction sums to zero.
第一，永远先画一张清晰的受力图，并分解成垂直的分量； 要平衡，每个方向的分量之和为零。
Second, use the suvat equations only when the acceleration is constant, and keep one consistent positive direction.
第二，只有加速度恒定时才用匀加速运动方程， 并保持一个一致的正方向。
Third, apply force equals mass times acceleration along the motion, including friction on a rough surface.
第三，沿运动方向应用力等于质量乘以加速度， 粗糙表面上还要算上摩擦力。
Fourth, state your assumptions — a light string, a smooth pulley, a particle — they are often worth a mark.
第四，写出你的假设——轻绳、光滑滑轮、质点——它们往往值一分。