Circular Motion
A-Level Physics Topic 12 13:00 English narration · English + 中文 subtitles burned in
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Fill a bucket with water, swing it in a fast circle over your head — and not a drop falls out, even when it is completely upside down.
往桶里装满水，抡起来在头顶上快速转圈——竟然一滴水都不会洒出来，哪怕它完全倒过来。
Why doesn't the water pour onto your face?
为什么水不会浇到你脸上？
Because the water is not trying to fall out — it is trying to fly off in a straight line.
因为水并不是想往下掉——它是想沿着直线飞出去。
The bucket is constantly pulling it inward, forcing it to curve.
桶不停地把它往里拉，迫使它拐弯。
That inward pull is a centripetal force, and it is the secret behind everything that moves in a circle.
这个向里的拉力就是向心力， 它是一切做圆周运动的东西背后的秘密。
Anything that goes round in a circle obeys the same rules.
任何绕圈运动的东西都遵守同样的规则。
Today: angles in radians, angular speed, and the force that bends a straight path into a circle.
今天：用弧度表示的角度、角速度， 以及那个把直线路径弯成圆的力。
Let's begin.
让我们开始吧。
First, a better way to measure angles: the radian.
首先，一种更好的量度角度的方法：弧度。
Take the radius of a circle, and bend it around the edge.
取圆的半径，把它沿着边缘弯过去。
The angle it sweeps out is one radian.
它所张开的角就是一弧度。
Go all the way around, and the full circle is two pi radians — the same as three hundred and sixty degrees.
绕一整圈，整个圆是二派弧度——等于三百六十度。
So a half turn is pi radians, and a quarter turn is pi over two.
所以半圈是派弧度，四分之一圈是二分之一派。
Radians make circular motion simple.
弧度让圆周运动变得简单。
Write it as a formula.
把它写成公式。
For an arc of length s on a circle of radius r, the angle in radians is theta equals s over r.
在半径为 r 的圆上，弧长为 s 时，用弧度表示的角等于 s 除以 r。
One radian is the angle whose arc length equals the radius.
一弧度就是弧长等于半径时所对的角。
Radians have no unit — they are just a ratio of two lengths.
弧度没有单位——它只是两个长度的比。
To convert: one radian is one hundred and eighty degrees over pi, about fifty-seven point three degrees.
换算：一弧度等于一百八十度除以派，大约五十七点三度。
Set your calculator to radians for this topic; degree mode will give wrong answers every time you press sine or cosine.
这一章请把计算器设成弧度制；如果开着角度制，一按正弦或余弦就会出错。
As the radius turns through an angle, the object moves along an arc.
当半径转过一个角时，物体沿圆弧移动。
That angle, measured in radians from a chosen start line, is the angular displacement, theta.
这个从选定起始线量起、用弧度表示的角， 就是角位移，西塔。
Angular speed, omega, is the rate of change of angular displacement.
角速度欧米伽是角位移的变化率。
For uniform motion omega is constant, so omega equals theta over t.
对匀速圆周运动，欧米伽是恒定的，所以欧米伽等于西塔除以 t。
The unit is radians per second.
单位是弧度每秒。
Watch the picture: as the radius sweeps through delta theta, the object travels an arc delta s at linear speed v, always along the tangent.
看图：当半径扫过德尔塔西塔时，物体以线速度 v 走过弧长德尔塔 s，速度始终沿切线。
How fast does something turn?
一个东西转得有多快？
We measure its angular speed — the angle it sweeps each second, called omega.
我们用它的角速度来量度——它每秒扫过的角度，叫做欧米伽。
In one full period, it turns through two pi radians, so omega is two pi divided by the period.
在一个完整的周期里，它转过二派弧度，所以欧米伽等于二派除以周期。
And there is a beautiful link to ordinary speed: the speed along the circle is the radius, times omega.
而它和普通速度之间有一个漂亮的联系：沿圆周的速度等于半径乘以欧米伽。
Twice as far out, and you move twice as fast, for the same rate of turning.
离中心远一倍，转动的快慢相同时，你走得也快一倍。
A spinning fairground ride turns every rider through the same angle each second.
旋转的游乐设施让每一位乘客每秒转过相同的角度。
If the object goes once round — two pi radians, one revolution — in time T, that time is the period.
如果物体转一整圈—— 二派弧度，一圈——所用时间是 T，那就是周期。
Then omega equals two pi over T.
于是欧米伽等于二派除以 T。
You can also write it as two pi times f, where f is the frequency of turning in hertz: how many revolutions each second.
也可以写成二派乘以 f，其中 f 是转动的频率，单位赫兹：每秒转多少圈。
Frequency is one over the period.
频率等于周期的倒数。
So three linked quantities: period T, frequency f, and angular speed omega.
所以有三个相互联系的量：周期 T、频率 f，和角速度欧米伽。
In one period the object travels a distance two pi r — the circumference — at constant speed, so v equals two pi r over T, which is the same as r times omega.
在一个周期里，物体走过的路程是二派 r——也就是周长——速率恒定， 所以 v 等于二派 r 除以 T，也就是 r 乘以欧米伽。
That links linear, or tangential, speed v with angular speed omega.
这就把线速度——也叫切向速度——v 和角速度欧米伽联系起来了。
Look at the turntable: both riders share the same angular speed, but the outer rider has the longer velocity arrow.
看转盘：两位乘客角速度相同，但外侧的速度箭头更长。
At a larger radius, for the same rate of turning, the linear speed is larger.
在相同的转动快慢下，半径越大，线速度越大。
A child on the edge of a merry-go-round moves faster than one near the centre, even though both go round once in the same time.
旋转木马边缘的孩子比靠近中心的孩子走得更快，尽管两人同一时间都转一圈。
A fairground ride of radius of four point zero metres completes one turn every eight point zero seconds.
一个游乐设施半径四米整，每八秒整转一圈。
Find its angular speed and the linear speed of a rider on the edge.
求它的角速度， 以及边缘乘客的线速度。
First, omega equals two pi over T: two pi over eight point zero is zero point seven nine radians per second.
首先，欧米伽等于二派除以 T： 二派除以八点零，是零点七九弧度每秒。
Then v equals r omega: four point zero times zero point seven nine is three point one metres per second.
然后 v 等于 r 乘以欧米伽： 四点零乘以零点七九，是三点一米每秒。
Same omega for every rider; larger r means larger v.
每位乘客的欧米伽相同；r 越大，v 越大。
Here is the strange thing about circular motion.
圆周运动有一件奇怪的事。
Even when something moves at a constant speed, it is always accelerating.
即使一个东西以恒定的速率运动，它也一直在加速。
How?
怎么会呢？
Because velocity is not just speed — it also has a direction.
因为速度不只是速率——它还有方向。
And the direction is changing, every instant.
而方向每一瞬间都在改变。
Watch the velocity arrow: its length never changes, but it is forever turning, always pointing along the circle, never toward the centre.
看那个速度箭头： 它的长度从不改变，但它永远在转动，始终沿着圆周，从不指向中心。
So a changing direction means an acceleration.
所以方向的改变意味着加速度。
But which way does it point?
但它指向哪个方向？
Straight toward the centre of the circle.
径直指向圆的中心。
We call it the centripetal acceleration — centre-seeking.
我们叫它向心加速度——指向中心。
And by Newton's second law, an acceleration needs a force in the same direction.
而根据牛顿第二定律，加速度需要一个同方向的力。
So there must be a force pulling the object inward, toward the centre: the centripetal force.
所以一定有一个力把物体向里拉，指向中心：这就是向心力。
Remove it, and the object flies off in a straight line.
去掉它，物体就沿直线飞出去了。
Look carefully at the directions.
仔细看方向。
The velocity always points along the tangent — never toward the centre.
速度始终沿切线——从不指向中心。
The force and the acceleration both point inwards, toward the centre.
力和加速度都指向内侧、指向中心。
The centripetal acceleration is perpendicular to the velocity at every instant — never along the direction of motion.
向心加速度在每一瞬间都垂直于速度——从不沿着运动方向。
If any part of it were along the motion, the speed would change.
如果它有任何分量沿运动方向，速率就会改变。
In uniform circular motion the speed stays constant, so the acceleration can only bend the path, not speed the object up or slow it down.
在匀速圆周运动中速率保持恒定， 所以加速度只能弯折路径，不能让物体加速或减速。
The unit is metres per second squared.
单位是米每二次方秒。
Now the equations.
现在来看方程。
The centripetal acceleration can be written two ways: the speed squared, divided by the radius, or the radius times omega squared.
向心加速度可以写成两种形式：速率的平方除以半径，或者半径乘以欧米伽的平方。
Multiply by the mass, and you get the centripetal force: mass times speed squared over radius, or mass times radius times omega squared.
再乘以质量，你就得到向心力：质量乘以速率平方除以半径，或者质量乘以半径乘以欧米伽的平方。
Faster motion, or a tighter circle, both demand a bigger force.
运动更快，或者圆更小，都要求更大的力。
The two forms are equal because v equals r omega.
两种形式相等，因为 v 等于 r 乘以欧米伽。
Substitute and they become the same expression.
代进去它们就变成同一个表达式。
Pick the one with the quantities you already have.
选你已经有的量对应的那一种。
Given speed and radius, use a equals v squared over r.
已知速率和半径，用 a 等于 v 平方除以 r。
Given omega and radius, use a equals r omega squared.
已知欧米伽和半径，用 a 等于 r 乘以欧米伽的平方。
And if a question mixes them, use v equals r omega to switch from one to the other.
如果题目混用，就用 v 等于 r 乘以欧米伽在两者之间切换。
A Ferris wheel: a centripetal force toward the centre keeps each car moving in a circle.
一座摩天轮：指向中心的向心力让每一节车厢沿圆周运动。
By Newton's second law, the resultant force on a body in circular motion at constant speed is F equals m a, which is m v squared over r, or m r omega squared.
根据牛顿第二定律， 在恒定速率圆周运动中，物体所受的合力等于 m a，也就是 m v 平方除以 r， 或者 m r 乘以欧米伽的平方。
This is the centripetal force.
这就是向心力。
It always points toward the centre — perpendicular to the velocity.
它始终指向中心——垂直于速度。
The key idea: centripetal force is not a new kind of force.
关键的一点：向心力不是一种新的力。
It is the net result of the real forces acting — tension, gravity, friction, electric attraction, normal contact force, and so on.
它是真实作用力的净效果—— 张力、重力、摩擦力、静电引力、支持力，等等。
Let's use it.
我们来用一用。
A ball of mass zero point two kilograms is swung on a string of length zero point five metres, at four turns per second.
一个质量零点二千克的球，系在一条零点五米长的绳子上， 以每秒四圈的速度旋转。
Find the force in the string.
求绳中的力。
First, the angular speed: four turns a second, times two pi, is about twenty-five radians per second.
首先，角速度：每秒四圈，乘以二派， 约为每秒二十五弧度。
Then the force is mass, times radius, times omega squared — zero point two, times zero point five, times twenty-five squared — about sixty-three newtons.
然后，力等于质量乘以半径再乘以欧米伽的平方—— 零点二，乘以零点五，乘以二十五的平方——约为六十三牛顿。
Same ball, different numbers.
同一个球，换一组数。
A zero point two zero kilogram ball on a string is whirled in a horizontal circle of radius zero point five zero metres at three point zero metres per second.
零点二零千克的球系在绳子上，在半径零点五零米的水平圆上， 以三点零米每秒旋转。
Find the centripetal force — the tension in the string.
求向心力——也就是绳中的张力。
This time we have v, so F equals m v squared over r: zero point two zero times three squared, over zero point five zero, equals three point six newtons.
这次我们有 v， 所以 F 等于 m v 平方除以 r：零点二零乘以三的平方，除以零点五零， 等于三点六牛顿。
Same idea, other form of the equation.
同样的思路，换用方程的另一种形式。
Always state what provides the centripetal force.
永远要说明是什么提供向心力。
A ball on a string in a horizontal circle: the tension in the string.
水平圆上的球和绳子：绳中的张力。
A car turning a flat corner: the friction between tyres and road — F equals m v squared over r.
汽车在平坦弯道转弯：轮胎与路面之间的摩擦力——F 等于 m v 平方除以 r。
If the car goes too fast, friction is not enough and it skids outwards.
车开得太快，摩擦力不够，就会向外滑出去。
A planet or satellite in orbit: the gravitational attraction, G M m over r squared equals m v squared over r.
行星或卫星在轨道上：引力， G M m 除以 r 平方等于 m v 平方除以 r。
An electron in a circular orbit around a nucleus: the electrostatic attraction between the electron and the positive nucleus equals m_e v squared over r.
电子绕原子核做圆周运动： 电子与带正电原子核之间的静电引力等于电子质量乘以 v 平方除以 r。
On a banked corner with no friction, the horizontal part of the normal contact force provides the centripetal force.
在没有摩擦的倾斜弯道上，支持力的水平分量提供向心力。
The road pushes the car at right angles to the surface.
路面垂直于表面推着汽车。
Resolve that normal force: the vertical part balances the weight, and the horizontal part points to the centre of the circle.
把这个支持力分解：竖直分量平衡重力， 水平分量指向圆心。
For the angle that needs no friction, tan theta equals v squared over r g.
对不需要摩擦的那个倾角，tan 西塔等于 v 平方除以 r g。
Steeper bank, or larger radius, and you can take the bend faster without relying on grip.
倾角更陡，或者半径更大，就可以在不依赖抓地力的情况下更快过弯。
When the circle is upright, the speed is not constant — gravity does work.
当圆是竖直的时候，速率并不恒定——重力在做功。
But at each instant the net force towards the centre still equals m v squared over r.
但每一瞬间， 指向中心的净力仍然等于 m v 平方除以 r。
At the bottom of a loop: tension up, weight down, so T minus m g equals m v squared over r — the tension is largest here.
在环的底部：张力向上，重力向下， 所以 T 减 m g 等于 m v 平方除以 r——这里张力最大。
At the top of a loop: tension and weight both point down, toward the centre, so T plus m g equals m v squared over r — the tension is smallest.
在环的顶部：张力和重力都向下、指向中心，所以 T 加 m g 等于 m v 平方除以 r—— 这里张力最小。
For the slowest speed at the top with the string just tight, set T equal to zero: m g equals m v_min squared over r, giving v_min equals the square root of g r.
顶部刚好绳子绷紧的最慢速度，令 T 等于零： m g 等于 m 乘以 v 最小的平方除以 r，得到 v 最小等于 g r 的平方根。
A car goes round a vertical loop of radius of two point zero metres.
一辆汽车绕半径两米整的竖直环行驶。
Find the minimum speed at the top for the car to keep contact with the track.
求顶部保持与轨道接触的最小速度。
Take g as nine point eight one.
取 g 为九点八一。
At the slowest speed the track force is zero, so gravity alone provides the centripetal force: m g equals m v min squared over r.
在最慢速度时轨道力为零，所以重力单独提供向心力： m g 等于 m 乘以 v 最小的平方除以 r。
Cancel m, and v min equals the square root of g r.
约掉 m，v 最小等于 g r 的平方根。
Square root of nine point eight one times two point zero is about four point four metres per second.
九点八一乘以两点零再开平方根，约为四点四米每秒。
One careful note.
有一点要小心。
The constant-speed results — v equals r omega, and omega constant — hold for horizontal circles, or where the force only bends the path.
恒定速率的结果——v 等于 r 乘以欧米伽，以及欧米伽恒定—— 适用于水平圆，或者力只弯折路径的情形。
That includes orbits in gravity and charges in a magnetic field.
这包括引力轨道和磁场中的电荷。
Vertical circles are different: gravity does work, so the speed changes as the object climbs and falls.
竖直圆不同：重力做功，所以物体上升和下降时速率会改变。
Always ask: is the force doing work, or only changing the direction?
永远要问：这个力是在做功，还是只在改变方向？
How to structure a circular-motion answer.
圆周运动答案怎么组织。
First, find the radius r and choose v or omega.
第一，找出半径 r，并选定用 v 还是欧米伽。
Use v equals r omega to switch between them.
用 v 等于 r 乘以欧米伽在两者之间切换。
Second, find the centripetal acceleration with a equals v squared over r, or r omega squared.
第二，用 a 等于 v 平方除以 r， 或者 r 乘以欧米伽的平方，求向心加速度。
Third, list the real forces and write Newton's second law in the radial direction — towards the centre is positive.
第三，列出真实的力， 并在径向——指向中心为正——写下牛顿第二定律。
Set the net inward force equal to m v squared over r.
令向内的净力等于 m v 平方除以 r。
Fourth, for period or frequency, use omega equals two pi over T, or T equals two pi r over v.
第四，若要求周期或频率， 用欧米伽等于二派除以 T，或者 T 等于二派 r 除以 v。
Finally, check the directions: centripetal force and acceleration point to the centre; the velocity is along the tangent.
最后检查方向： 向心力和向心加速度指向中心；速度沿切线。
Three marks to secure.
三个要拿稳的分。
First, always work in radians, not degrees, when you use omega.
第一，凡是用到欧米伽，就一定用弧度，而不是角度。
Second, the centripetal force points toward the centre — it is not a new force, but whatever provides that pull: gravity, tension, or friction.
第二，向心力指向中心——它不是一个新的力，而是提供这个拉力的东西：重力、张力或摩擦力。
Third, at constant speed the force does no work, because it is always perpendicular to the motion.
第三，在恒定速率下，这个力不做功，因为它始终垂直于运动方向。
Master these, and circular motion is yours.
掌握这些，圆周运动就是你的了。