Force & Translational Dynamics
AP Physics 1 Topic 2 10:50 English narration · English + 中文 subtitles burned in
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Three skydivers, far above the ground.
三名跳伞者，在离地面很高的空中。
Gravity is pulling every one of them down, hard.
重力正用力把他们每个人往下拉。
Yet they hang there, steady, side by side.
可是他们并排悬在那里，稳稳的。
So why has that pull stopped mattering?
那么，为什么那股拉力好像不起作用了？
The answer is about forces.
答案在于力。
This unit teaches you to find every force acting on a thing, add them up, and turn that sum into an acceleration.
本单元教你找出作用在物体上的每一个力，把它们相加， 再把这个总和变成加速度。
That one move runs through everything on this list.
这一步贯穿了这张清单上的每一项。
First, choose your system: the object, or group of objects, you decide to study.
先选定你的系统：你决定研究的那个物体，或那一组物体。
Then treat the whole thing as one point — its centre of mass.
然后把整体当成一个点来处理——它的质心。
It is the average position of the mass, so on this rod it sits closer to the heavier ball.
质心是质量的平均位置， 所以在这根杆上，它更靠近较重的那个球。
Outside forces move that point; internal ones never do.
外力会推动这个点；内力永远不会。
A force is a push or a pull, measured in newtons, and it always comes from another object — nothing pushes itself.
力是推或拉，单位是牛顿，而且它总是来自另一个物体——没有东西能推自己。
So before you calculate, draw a free-body diagram.
所以在计算之前，先画受力图。
Here the surface pushes up, the Earth pulls down, a hand pulls right, friction drags left.
这里，地面向上推，地球向下拉， 一只手向右拉，摩擦力向左拖。
Four named arrows.
四个带名字的箭头。
Two rules keep that diagram honest.
有两条规则让这张图保持诚实。
The first rule: draw only the forces acting on this object, and shrink it to a dot.
第一条规则：只画作用在这个物体上的力， 并把它缩成一个点。
So the block's push on the floor never appears — that belongs to the floor's diagram.
所以物块对地面的压力绝不出现——那属于地面自己的受力图。
Second, draw real, physical forces only: a hand, a rope, a surface, gravity.
第二，只画真实的力：手、绳、接触面、重力。
And never draw a mass-times-acceleration arrow.
绝不要画「质量乘加速度」的箭头。
That arrow is the result of the forces, not a force itself.
那个箭头是力的结果，不是力本身。
Last, aim one axis along the acceleration — on an incline, that means along the slope.
最后，让一个坐标轴沿着加速度的方向——在斜面上，就是沿着斜面。
Newton's third law.
牛顿第三定律。
If A pushes on B, then B pushes back on A, equal in size and opposite in direction.
如果甲推乙，乙就反过来推甲，大小相等，方向相反。
One arrow acts on B, the other on A.
一个箭头作用在乙上，另一个作用在甲上。
Different objects — so they can never cancel.
不同的物体——所以它们永远不会互相抵消。
That is why a rope works: one tension, all along it.
绳子能起作用也是这个道理：一个张力，贯穿全绳。
Here is the exam's favourite trap.
这是考试最爱的陷阱。
A book lies on a table.
一本书放在桌上。
Its weight pulls down, the table pushes up.
它的重力向下拉，桌子向上推。
Equal and opposite — so they look like a pair.
大小相等、方向相反——所以看起来像一对。
They are not.
其实不是。
Both act on the same object, the book.
它们都作用在同一个物体上，那本书。
The weight's true partner is the pull the book makes on the Earth.
重力真正的搭档，是书对地球的吸引。
The table's true partner is the push the book makes on the table.
桌子支持力真正的搭档，是书对桌子的压力。
So always ask: who pushes whom?
所以永远要问：是谁在推谁？
Name the two objects, every time.
每一次都把这两个物体说出来。
Back to those skydivers.
回到那几位跳伞者。
Early in the jump gravity wins and they speed up, but air resistance grows with speed.
跳下之初重力占上风，他们越落越快， 但空气阻力随速度增大。
Watch the two arrows become equal.
看这两个箭头逐渐相等。
Once they match, the net force is zero — translational equilibrium — and the first law says the velocity then stays as it is.
一旦相等，合力为零， 第一定律说速度就保持原样不变。
That is terminal velocity.
这就是收尾速度。
Zero net force means no change, not rest; the property that resists a change is inertia.
合力为零意味着「不变」，而不是「静止」。
When the forces do not balance, the object accelerates.
当各个力不平衡时，物体就会加速。
Newton's second law: acceleration equals the net force divided by the mass.
牛顿第二定律：加速度等于合力除以质量。
It points the same way as the net force, and the same force gives a heavier object less of it — that resistance is its inertial mass.
它的方向与合力相同，而同样的力作用在更重的物体上，加速度更小。
Same push on both carts: the light one speeds up quickly, the heavy one hardly moves.
两辆小车受到同样的推力：轻的很快加速，重的几乎不动。
Put a number through it.
给它代个数。
A four kilogram box is pulled along the floor by eighteen newtons, and friction on it is six newtons.
一个四千克的箱子被十八牛顿的力沿地面拉动，箱子受到的摩擦力是六牛顿。
Find its acceleration.
求它的加速度。
Motion is horizontal, so add them along that axis: eighteen forward, six back, net twelve newtons.
运动是水平的，所以就沿这个方向相加：向前十八，向后六，合力十二牛顿。
Then the second law says acceleration is the net force divided by the mass.
然后第二定律说，加速度等于合力除以质量。
So it is twelve divided by four, which is three metres per second squared, forward.
所以是十二除以四，等于每二次方秒三米，方向向前。
Notice the order — one axis at a time, net force first, divide last.
注意这个顺序——一次只处理一个方向，先求合力，最后再除。
That habit is what keeps a two-force problem and a six-force problem the same problem.
正是这个习惯，让两个力的题和六个力的题变成同一道题。
A block slides down a smooth ramp, tilted at thirty degrees. Find its acceleration.
一个物块沿着倾角三十度的光滑斜面下滑，求它的加速度。
First, draw the diagram: the weight straight down, the normal force at right angles to the surface.
第一步，画图：重力竖直向下，法向力垂直于斜面。
Next, choose the axes — one along the slope, since that is the way it accelerates.
第二步，选坐标轴——让一个轴沿着斜面，因为物块正是朝那个方向加速。
Now resolve the weight into those directions.
现在把重力分解到这两个方向上。
Along the slope, the part that acts is the weight times the sine of the angle.
沿斜面的那一部分等于重力乘以角的正弦。
Nothing balances it, so that is the net force.
没有力与它平衡，所以它就是合力。
Put it into the second law, and the mass cancels.
代入第二定律，质量被约掉。
So the answer is ten times the sine of thirty degrees: five metres per second squared.
于是答案是十乘以三十度的正弦：五米每二次方秒。
And there is no mass in the answer: a heavy block and a light one slide exactly alike.
答案里没有质量：重的物块和轻的物块下滑得完全一样。
Where does weight come from?
重力从哪里来？
Gravity.
来自万有引力。
Near a planet's surface the pull on an object is its mass times the gravitational field strength, straight down — and that pull is its weight.
在行星表面附近，作用在物体上的吸引力 等于它的质量乘以重力场强度，方向竖直向下——这个吸引力就是它的重力。
Near the Earth the field is about nine point eight newtons per kilogram; the exam lets you round it to ten.
在地球附近，场强约为每千克九点八牛顿；考试允许你取整为十。
It barely changes near the ground.
在地面附近，它几乎不变。
That difference has a number attached, and the exam checks it.
这个区别是有数字的，考试也会查。
Take a two kilogram object.
取一个两千克的物体。
On Earth the field is nine point eight newtons per kilogram, so its weight is two times nine point eight — nineteen point six newtons.
在地球上，场强是每千克九点八牛顿，所以它的重量是二乘九点八——十九点六牛顿。
Carry it to the Moon.
把它带到月球。
On the Moon the field is only one point six, so the weight becomes two times one point six, which is three point two newtons.
月球上的场强只有一点六， 所以重量变成二乘一点六，等于三点二牛顿。
But the mass never changed: it is two kilograms in both places.
但质量从来没有变：在两个地方都是两千克。
Mass measures inertia — how hard the object is to accelerate — and it travels with the object.
质量衡量的是惯性——物体有多难被加速——它随物体一起走。
Weight is a force, and it belongs to where the object is standing.
重量是一个力，它属于物体所站的地方。
Gravity is not only for planets.
引力不只属于行星。
Newton's law of gravitation: any two masses attract each other.
任意两个质量都会互相吸引。
The force grows with each mass and falls with the square of the distance between their centres: double the distance, and the pull drops to a quarter.
这个力随两者的质量增大，并随它们中心之间距离的平方而减小： 距离加倍，吸引力就降到四分之一。
The arrows are equal and opposite, along the line joining them — a third-law pair.
两个箭头大小相等、方向相反， 沿着连线——这也是第三定律的一对力。
On the Moon, a two-kilogram object keeps its mass but weighs far less.
在月球上，一个两千克的物体质量不变， 但重力小得多。
What does a bathroom scale really measure?
体重计究竟测的是什么？
Not your weight.
不是你的重力。
It measures the normal force — the push of the floor on you.
它测的是法向力——地板对你的支持力。
Stand still, or ride at a steady speed, and it equals your weight: six hundred newtons for a sixty-kilogram student.
站着不动，或者匀速上下，它就等于你的重力：六十千克的学生对应六百牛顿。
Now accelerate upward: the floor pushes harder, and the reading climbs.
现在向上加速：地板推得更用力，读数升高。
Accelerate downward and it pushes less.
向下加速，它推得更小。
And in free fall it does not push at all, so the scale reads zero.
而在自由落体中，它完全不推，所以读数为零。
You feel weightless, but your weight never changed.
你会觉得失重，但你的重力从未改变。
Friction.
摩擦力。
Push a heavy crate gently and nothing happens: friction pushes back with exactly the same size.
轻轻推一个重箱子，什么也不会发生：摩擦力以完全相同的大小顶回来。
That is static friction, and it adjusts to whatever is needed.
这就是静摩擦，它会自动调整到所需要的大小。
But it has a limit — the coefficient times the normal force.
但它有一个上限——摩擦系数乘以法向力。
Push past that peak and the crate breaks free; friction then settles at a lower, steady value: kinetic friction.
推过这个峰值，箱子就挣脱了； 摩擦力随后落到一个更小的恒定值：动摩擦。
Springs obey a rule just as simple — Hooke's law.
弹簧遵守的规则同样简单——胡克定律。
Hang equal weights on a spring and it stretches in equal steps.
挂上相等的重物，它就伸长相等的长度。
The force it pulls back with is proportional to how far you stretched or squashed it, and the number linking them is the spring constant.
它回拉的力与你把它拉长或压短的距离成正比， 而联系两者的那个数就是弹簧劲度系数。
It always points back toward the natural length, which is why it is called a restoring force.
它总是指回自然长度， 所以叫做回复力。
The minus sign matters: that force always points back to the relaxed position.
负号很重要： 这个力总是指回自然长度的位置。
One number for the spring too.
弹簧也来一个数。
A spring of stiffness two hundred newtons per metre hangs vertically, and a half kilogram mass is hung on it.
一根劲度系数为每米两百牛顿的弹簧竖直悬挂， 在它上面挂一个半千克的物体。
How far does it stretch when it settles?
它稳定后伸长多少？
At rest nothing accelerates, so the spring force balances the weight: k x equals m g.
静止时没有加速度，所以弹簧的力与重力平衡：k 乘 x 等于 m 乘 g。
Rearrange for the stretch — m g over k.
解出伸长量——m g 除以 k。
That is nought point five, times nine point eight, over two hundred, which is nought point zero two five metres, or two point five centimetres.
也就是零点五，乘九点八，再除以两百，等于零点零二五米，也就是二点五厘米。
And notice where that equation came from: balance, not a formula to memorise.
注意这个方程是从哪来的：从平衡来的，不是背来的公式。
Every hanging-spring question starts the same way.
每一道悬挂弹簧的题，都是这样开头的。
One last idea.
最后一个概念。
This ball moves at a constant speed, yet it accelerates the whole time, because its direction keeps changing.
这个球以恒定的速率运动，却始终在加速， 因为它的方向一直在改变。
The velocity lies along the tangent; the acceleration points at the centre, and its size is the speed squared divided by the radius.
速度沿着切线；加速度指向圆心， 大小等于速率的平方除以半径。
Any real inward force supplies it — a string, gravity, friction.
任何指向圆心的真实的力都能提供它—— 绳、引力、摩擦力。
One turn takes the period; the turns per second are the frequency.
转一圈所用的时间叫周期；每秒转过的圈数叫频率。
One classic question.
来一道经典题。
A three-hundred-gram ball on a string is whirled in a circle of radius zero point eight metres, at four metres per second.
一个三百克的小球系在绳上，在半径零点八米的圆周上， 以四米每秒转动。
Find the tension.
求绳中的张力。
Pause here and try it.
先暂停，自己试一试。
First, name the inward force: the string alone pulls the ball inward, so the tension is the whole centripetal force.
第一步，指出向心的那个力：只有绳把球拉向圆心，所以张力就是全部的向心力。
Then write the rule — mass times speed squared, over the radius.
然后写出规则——质量乘以速率的平方，再除以半径。
Put the numbers in.
代入数字。
That gives a tension of six newtons.
得到张力为六牛顿。
So if the string can hold only six newtons, that is the ball's top speed.
所以如果绳最多只能承受六牛顿，那就是球的最大速率。
Now use that limit to decide a case, because that is how it is asked.
现在用这个上限去判断一个具体情形，因为考试就是这样问的。
A five kilogram crate sits on a level floor, and the coefficient of static friction is nought point four.
一个五千克的箱子放在水平地面上，静摩擦系数是零点四。
Will a horizontal push of fifteen newtons move it?
十五牛顿的水平推力能推动它吗？
On a level floor the normal force is the weight: five times nine point eight, forty-nine newtons.
在水平地面上，法向力等于重力：五乘九点八，四十九牛顿。
So the largest static friction available is nought point four times forty-nine — nineteen point six newtons.
所以能提供的最大静摩擦力是零点四乘四十九——十九点六牛顿。
Fifteen is less than nineteen point six, so the crate does not move.
十五小于十九点六，所以箱子不动。
And read the last part carefully: friction right now is fifteen newtons, not nineteen point six.
最后一点要读仔细：此刻的摩擦力是十五牛顿，不是十九点六。
Static friction only ever matches the push, up to its limit — it does not jump to the maximum just because the maximum exists.
静摩擦力永远只与推力相等，直到它的上限—— 它不会因为上限存在，就直接跳到最大值。
Four marks students give away every year.
四个学生年年送出去的分数。
First, a third-law pair acts on two different objects — a book's weight and the table's push are not a pair.
第一，第三定律的一对力作用在两个不同的物体上—— 书的重力和桌子的支持力不是一对。
Second, work one axis at a time; on a slope, put an axis along the slope.
第二，一次只处理一个坐标轴； 在斜面上，就让一个轴沿着斜面。
Third, static friction adjusts: at most the coefficient times the normal force.
第三，静摩擦会自动调整： 最大等于摩擦系数乘以法向力。
Fourth, there is no outward force in circular motion.
第四，圆周运动中没有向外的力。
Get those four, and this unit is yours.
把这四点做对，这个单元就是你的了。