Moments & Pressure
A-Level Physics Topic 4 17:06 English narration · English + 中文 subtitles burned in
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The Leaning Tower of Pisa has tilted for over eight hundred years — yet it does not fall.
比萨斜塔已经倾斜了八百多年——却始终不倒。
Why?
为什么？
The whole weight of the tower acts at one point: its centre of gravity.
整座塔的重量都作用在一个点上： 它的重心。
Drop a straight line down from that point.
从那一点向下画一条竖直线。
As long as the line stays inside the tower's base, the tower stands.
只要这条线还落在塔的底座之内，塔就能站立。
Lean it just a little further, and the line would fall outside the base — and gravity would pull it over.
再倾斜一点点，这条线就会落到底座之外——重力就会把它拉倒。
This is the physics of turning and balance.
这是关于转动与平衡的物理学。
Today: moments and torque, the rule of equilibrium, and then density, pressure, and why a steel ship floats.
今天：力矩与力偶矩、平衡的法则， 然后是密度、压强，以及为什么一艘钢铁大船能浮起来。
Let's begin.
让我们开始吧。
Every object's weight can be treated as acting at one single point — its centre of gravity.
每个物体的重量都可以看作作用在一个点上——它的重心。
For a uniform, regular shape, it sits right at the middle.
对于均匀、规则的形状， 重心就在正中间。
On a free-body diagram, always draw the weight arrow starting there.
在受力图上，重力箭头一定从那里画起。
And here is the rule for toppling: an object stays upright as long as the line straight down from its centre of gravity lands inside its base.
这里有一条关于倾倒的规则： 只要从重心竖直向下的那条线落在底座之内，物体就保持竖立。
For an irregular shape you find it by experiment.
形状不规则的物体要靠实验来找。
Hang the object freely from a pin: in equilibrium it settles with its centre of gravity vertically below the pin, because only then does the weight have no moment about the pin.
把物体用一根针自由地挂起来：平衡时它会停在重心正好在针的正下方的位置， 因为只有这样，重力对针才没有力矩。
A plumb line hung from the same pin marks that vertical, so you draw it on.
从同一根针挂一条铅垂线，就标出了这条竖直线，把它画在物体上。
Repeat from a second point, and the centre of gravity is where the two lines cross.
换一个悬挂点再做一次，两条线的交点就是重心。
To turn something, you apply a moment.
要让东西转动，你就施加一个力矩。
The moment of a force equals the force times the perpendicular distance to the pivot.
力对某点的力矩，等于力乘以到支点的垂直距离。
The further out you push, the bigger the turning effect — which is why a long spanner loosens a tight nut.
你推得越远，转动效果越大——这就是为什么长扳手能拧开很紧的螺母。
If the force acts at an angle, only the perpendicular part turns it, so the moment is the force, times the arm, times the sine of the angle.
如果力是斜着作用的，只有垂直的那部分让它转动，所以力矩等于力乘以力臂再乘以角度的正弦。
Fifty newtons, on a spanner forty centimetres long, at thirty degrees, gives ten newton metres.
五十牛顿，作用在四十厘米长的扳手上，成三十度角，得到十牛顿米。
Look carefully at the diagram.
仔细看这张图。
A force F acts along a line that does not pass through the pivot.
力沿一条不经过支点的作用线。
The moment is not the force times the arm length alone.
力矩不是力乘以力臂长度那么简单。
It is the force times the perpendicular distance from the pivot to that line of action.
它是力乘以从支点到作用线的垂直距离。
On the diagram, that perpendicular distance is the dashed length marked d.
图上那条虚线标出的，就是这个垂直距离。
A moment about a point is either clockwise or anticlockwise — always state which way it turns.
对某一点的力矩要么是顺时针，要么是逆时针——一定要说明转向。
The unit is the newton metre.
单位是牛顿米。
Here is the same idea with an angle.
同样的思路，但这次有角度。
The force acts at angle theta to the lever arm of length r.
力与长度为力臂的方向成某个角。
Only the part of the force perpendicular to the arm produces a turn — that component is F sine theta.
只有垂直于力臂的那部分力才产生转动—— 那个分量是力乘以角的正弦。
So the moment is F times r times sine of theta.
所以力矩等于力乘以力臂再乘以角的正弦。
Work the numbers: fifty newtons, arm zero point four zero metres, thirty degrees.
算一下：五十牛顿，力臂零点四零米，三十度。
Sine of thirty is one half, so the moment is ten newton metres.
三十度的正弦是二分之一，所以力矩是十牛顿米。
Convert centimetres to metres before you multiply.
相乘之前先把厘米换成米。
Two equal forces, pointing opposite ways, a distance apart, form a couple.
两个大小相等、方向相反、相隔一段距离的力，构成一个力偶。
A couple has no resultant force, so it does not push the object anywhere — it only makes it turn.
力偶没有合力， 所以它不会把物体推向任何地方——它只让物体转动。
Its turning effect is the torque: one force, times the distance between the two lines.
它的转动效果就是力偶矩： 一个力乘以两条作用线之间的距离。
Watch out for a trap: two forces pointing the same way are not a couple.
当心一个陷阱：两个方向相同的力不是力偶。
A couple needs equal size and opposite direction.
力偶需要大小相等且方向相反。
Study the couple on the disc.
看圆盘上的力偶。
Two equal forces F point opposite ways, separated by the diameter.
两个大小相等的力方向相反，相隔一个直径。
There is no net force, so the centre of the disc is not pushed sideways — the disc only turns.
没有净力， 所以圆盘中心不会被推偏——圆盘只转动。
The torque equals one force times the perpendicular separation of the two lines of action.
力偶矩等于一个力乘以两条作用线之间的垂直间距。
A special property: the torque of a couple is the same about every point in the plane.
有一个特殊性质：力偶的力矩对平面内任意一点都相同。
That is why couples are so useful for describing pure rotation.
这就是为什么力偶特别适合描述纯转动。
Another view of a couple.
再看力偶的另一种画法。
One force up, one force down, separation d between them.
一个力向上，一个力向下，两者相隔距离。
The turning effect is F times d.
转动效果等于力乘以这个间距。
If both arrows pointed the same way, you would have a resultant force and the object would accelerate in a straight line — translation, not a couple.
如果两个箭头方向相同，就会有合力，物体会直线加速——那是平动，不是力偶。
Remember: equal size and opposite direction, with a gap between the lines.
记住：大小相等、方向相反，并且两条作用线之间要有间隔。
That gap is what makes the torque.
那个间隔才产生力偶矩。
The definition to write for the torque of a couple: the product of one of the forces and the perpendicular distance between the lines of action of the forces.
力偶矩要写的定义是： 其中一个力，与两个力的作用线之间的垂直距离的乘积。
One of the forces, not both, and the perpendicular distance between the two lines — not the distance to any pivot, because a couple has no pivot to measure from.
是其中一个力，不是两个加起来； 而且是两条作用线之间的垂直距离——不是到某个支点的距离， 因为力偶根本没有支点可以量。
An object is in equilibrium when two things are true at once.
当两件事同时成立时，物体处于平衡。
First, the resultant force is zero — no straight-line acceleration.
第一，合力为零——没有直线方向的加速度。
Second, the resultant moment about any point is zero — no angular acceleration, no turning.
第二，合力矩为零——没有转动。
You need both.
两者都需要。
An object can have zero resultant force and still be spinning; it can have zero resultant moment and still be sliding.
一个物体可以合力为零却仍在旋转； 也可以合力矩为零却仍在滑动。
Only when both are zero is it truly in balance.
只有当两者都为零时，它才真正处于平衡。
For a balanced object, there is a beautifully simple rule.
对于平衡的物体，有一条极其简单的规则。
The total clockwise moment about any point equals the total anticlockwise moment about that same point.
绕任意一点的总顺时针力矩， 等于绕同一点的总逆时针力矩。
This is the principle of moments.
这就是力矩原理。
Watch the beam.
看这根横梁。
Add weight on one side, and the clockwise moment grows; to rebalance, the anticlockwise moment must grow to match.
在一侧加上重物，顺时针力矩就增大；要重新平衡，逆时针力矩就必须随之增大。
When they are equal, the beam stays level.
当两者相等时，横梁保持水平。
Here is the principle of moments written under a simple beam. Weight W one times distance d one equals weight W two times distance d two, when the beam is balanced.
力矩原理写在一根简单横梁下面：一侧重量乘以距离，等于另一侧重量乘以距离，横梁才平衡。
Clockwise total equals anticlockwise total about the pivot.
绕支点的总顺时针等于总逆时针。
Use this whenever a rod, plank, or see-saw is not turning.
只要杆、板或跷跷板不在转动，就可以用这条规则。
If a uniform beam is heavy, put its weight at its centre of gravity and include that moment too, unless the centre sits on the pivot itself.
如果均匀横梁本身很重，要把重量放在重心处并计入力矩——除非重心正好落在支点上。
To solve a balance problem, follow four steps.
解平衡问题，按四步走。
First, choose a pivot — usually where an unknown force acts, so that force has zero distance and drops out.
第一，选支点——通常选未知力所在处，这样该力距离为零就从方程里消掉。
Second, list every force and its perpendicular distance from that pivot.
第二，列出每一个力及其到支点的垂直距离。
Third, set the total clockwise moment equal to the total anticlockwise moment.
第三，令总顺时针力矩等于总逆时针力矩。
Fourth, if you still need another unknown, use the force condition as well: the vector sum of all forces is zero.
第四，若还差一个未知量，再用力的条件：所有力的矢量和为零。
For a heavy uniform rod, never forget its weight at the centre of gravity.
对于很重的均匀杆，千万别忘了重心处的重量。
Let's use it.
我们来用一用。
A uniform beam is pivoted at its centre.
一根均匀横梁在中心处被支起。
A forty-newton weight hangs one and a half metres to the left.
一个四十牛顿的重物挂在左侧一点五米处。
How far to the right must a thirty-newton weight hang to balance it?
一个三十牛顿的重物要挂在右侧多远处才能使它平衡？
The beam's own weight sits on the pivot, so it makes no moment.
横梁自身的重量落在支点上， 所以不产生力矩。
Set clockwise equal to anticlockwise: forty times one point five, equals thirty times the distance.
让顺时针等于逆时针：四十乘以一点五，等于三十乘以距离。
So the distance is two metres.
所以这个距离是两米。
Watch for the word non-uniform.
要留意「非均匀」这个词。
A uniform beam has its weight at the centre; a non-uniform one does not, and the question will either give you where its centre of gravity is or make finding it the thing being asked.
均匀的横梁重心在正中间；非均匀的不在， 题目要么会告诉你它的重心在哪，要么就是让你把它求出来。
Put the weight arrow at that point, not at the middle out of habit.
把重力的箭头画在那个位置，不要习惯性地画在中点。
Look at the worked beam again.
再看一遍这道横梁例题。
Forty newtons hangs one point five metres left of the central pivot.
四十牛顿挂在中心支点左侧一点五米处。
Thirty newtons hangs a distance d to the right.
三十牛顿挂在右侧距离处。
The beam is uniform, so its own weight acts at the centre — on the pivot — and makes no moment.
横梁均匀，自身重量作用在中心——也就是支点上——不产生力矩。
Set clockwise equal to anticlockwise: forty times one point five equals thirty times d.
令顺时针等于逆时针：四十乘以一点五等于三十乘以距离。
Sixty equals thirty d, so d is two point zero metres.
六十等于三十乘距离，所以距离是两点零米。
Always state the unit.
一定要写出单位。
When just three forces hold an object in equilibrium, there is an elegant shortcut.
当只有三个力使物体保持平衡时，有一个优雅的捷径。
Draw the three forces tip to tail.
把这三个力首尾相接地画出来。
Because they balance, the arrows close into a triangle — the end meets the start.
因为它们相互平衡，这些箭头就闭合成一个三角形——终点回到起点。
From that closed triangle, you can find any unknown force with simple trigonometry, without splitting anything into components.
从这个闭合三角形，你就能用简单的三角函数求出任何未知的力，而不必把任何东西分解成分量。
Three coplanar forces in equilibrium form a closed vector triangle when drawn tip to tail.
三个共面且平衡的力，首尾相接画出来会形成闭合矢量三角形。
On the left you see weight W and two tensions meeting at a point.
左边是重量与两个张力交于一点。
On the right those three arrows close into a triangle.
右边那三个箭头闭合成三角形。
From the closed triangle you can find an unknown size or direction with the sine rule or the cosine rule, or by drawing the triangle carefully to scale.
从闭合三角形，你可以用正弦定理或余弦定理， 或按比例仔细作图，求出未知的大小或方向。
You can also split each force into horizontal and vertical components and set the sum of each direction to zero — same answer, different method.
也可以把每个力分解成水平与竖直分量， 令每个方向的代数和为零——答案相同，方法不同。
Now, matter and fluids.
现在来讲物质和流体。
Density is how much mass is packed into each unit of volume — mass divided by volume.
密度是每单位体积里装了多少质量——质量除以体积。
Water is one thousand kilograms per cubic metre.
水是每立方米一千千克。
Iron is nearly eight times denser.
铁的密度差不多是它的八倍。
Air is nearly a thousand times less.
空气则差不多小一千倍。
Density decides whether things float or sink: ice is a touch less dense than water, so an iceberg floats — with most of itself hidden below the surface.
密度决定东西是浮还是沉：冰的密度比水略小，所以冰山会浮起来——大部分藏在水面之下。
This iceberg shows density at work.
这座冰山展示了密度的作用。
Density is mass divided by volume — a scalar, so it has size but no direction.
密度是质量除以体积——它是标量，只有大小没有方向。
Water is one thousand kilograms per cubic metre, which is also one gram per cubic centimetre.
水是每立方米一千千克，也就是每立方厘米一克。
Iron and steel are about seven thousand eight hundred.
铁和钢大约是七千八百。
Air at room conditions is only about one point two.
室温下的空气大约只有一点二。
Ice is a little less dense than liquid water, so the berg floats with most of its volume hidden below the water line.
冰比液态水密度略小，所以冰山浮着，大部分体积藏在水线以下。
That fraction depends on the density ratio.
这个比例取决于密度比。
Pressure is force per unit area, with the force acting at right angles to the surface.
压强是单位面积上的力，且力要垂直于表面。
The unit is the pascal — one newton per square metre.
单位是帕斯卡——每平方米一牛顿。
This aneroid barometer measures atmospheric pressure in hectopascals, which are hundreds of pascals.
这台空盒气压计以百帕斯卡量度大气压强，也就是一百帕斯卡为一格。
Sea-level atmospheric pressure is about one times ten to the five pascals — roughly one hundred thousand newtons on every square metre.
海平面大气压大约是一乘以十的五次方帕斯卡——大约每平方米十万牛顿。
Fluids push in all directions; the size of that push at a depth is what we call fluid pressure.
流体向各个方向推；在某一深度上这个推力的大小，就是流体压强。
Pressure is force spread over an area — force divided by area, measured in pascals.
压强是力分摊在一块面积上——力除以面积，用帕斯卡来量度。
In a fluid, pressure grows with depth.
在流体中，压强随深度增大。
Picture a column of liquid: its own weight presses on the area below it.
想象一柱液体：它自身的重量压在下方的面积上。
Work it through, and the extra pressure equals the density, times gravity, times the depth.
推导一遍， 额外的压强就等于密度乘以重力加速度再乘以深度。
It depends only on depth, not on the shape of the container.
它只取决于深度，而与容器的形状无关。
Two and a half metres of water adds about twenty-five thousand pascals.
两米半的水会增加大约两万五千帕斯卡。
Derive the hydrostatic equation from a liquid column.
从一柱液体导出流体静压强方程。
A column of fluid has density rho, cross-sectional area A, and height delta h.
一柱流体有密度、横截面积和高度差。
Its mass is density times volume, so mass is rho times A times delta h.
质量等于密度乘以体积，也就是密度乘以面积再乘以高度差。
Its weight is that mass times g.
重量是质量乘以重力加速度。
That weight presses on the base area A, so the extra pressure at the bottom is weight over A — which simplifies to rho g delta h.
这个重量压在底部面积上，所以底部多出的压强是重量除以面积——化简后就是密度乘重力加速度乘高度差。
Only density and depth matter.
只与密度和深度有关。
The shape of the container cancels out.
容器的形状会消掉。
Write it cleanly: the extra pressure is rho g h.
写干净一点：额外压强等于密度乘重力加速度乘深度。
For two point five metres of fresh water, take density one thousand, g nine point eight one, times two point five.
两米半的淡水， 取密度一千、重力加速度九点八一，再乘两点五。
That is about two point five times ten to the four pascals — twenty-five thousand pascals.
大约是二点五乘以十的四次方帕斯卡——两万五千帕。
For a submarine at depth h, the water alone contributes density of seawater times g times h.
潜艇在深度处，仅水就贡献海水密度乘重力加速度乘深度。
The total pressure on the hull is that value plus the atmospheric pressure at the surface — about one times ten to the five pascals more.
船体上的总压强还要加上表面的大气压——大约再多一乘以十的五次方帕斯卡。
Finally, why things float.
最后，为什么东西会浮。
Push an object under water.
把一个物体按到水下。
The pressure on its bottom is greater than on its top, because the bottom is deeper.
它底部受到的压强比顶部更大，因为底部更深。
That difference pushes it upward — the upthrust.
这个差值把它向上推——这就是浮力。
Archimedes found its size: it equals the weight of the fluid the object pushes aside, the density, times gravity, times the volume.
阿基米德发现了它的大小：它等于物体所排开流体的重量， 也就是密度乘以重力加速度再乘以体积。
A block of two litres, held under water, feels about twenty newtons of upthrust.
一个两升的方块按在水下，受到约二十牛顿的浮力。
And a steel ship floats because it pushes aside enough water to match its whole weight.
而一艘钢铁大船能浮起来，是因为它排开的水足以抵消它的全部重量。
See why upthrust appears.
看浮力从何而来。
The top face of the block sits at depth h one; the bottom face sits at depth h two, which is deeper.
方块顶面在较浅的深度，底面在更深的深度。
Pressure at the bottom is therefore greater.
所以底部压强更大。
The downward force on the top is pressure times area; the upward force on the bottom is larger.
顶面的向下力是压强乘以面积；底面的向上力更大。
Their difference is the net upthrust.
两者之差就是净浮力。
Sideways forces cancel.
侧面的力互相抵消。
For an incompressible fluid, the upthrust depends on fluid density and displaced volume — not on how deep the whole object sits, and not on what the object is made of.
对于不可压缩流体，浮力取决于流体密度和排开体积——与整体沉多深无关，也与物体材料无关。
A floating object sinks until the upthrust on the submerged part equals its weight.
漂浮的物体会下沉，直到浸没部分所受浮力等于自身重量。
Fully submerged, the displaced volume is the object's full volume.
完全浸没时，排开体积就是物体的全部体积。
Partly submerged, only the volume below the water line counts.
部分浸没时，只算水线以下的体积。
Set upthrust equal to weight: density of fluid times g times submerged volume equals the object's weight.
令浮力等于重量：流体密度乘重力加速度乘浸没体积等于物体重量。
That is why a denser fluid supports the same object with less of it under the surface — less volume is needed to match the weight.
所以密度更大的流体可以用更少的浸没体积托住同一物体——匹配重量所需体积更小。
Two useful force balances.
两个有用的受力平衡。
A block held under water by a string tied to the floor feels three vertical forces: weight down, tension down, and upthrust up.
用绳子拴在池底、按在水下的方块受三个竖直力：重力向下、张力向下、浮力向上。
In equilibrium, upthrust equals weight plus tension, so you can solve for the tension.
平衡时浮力等于重量加张力，于是可以解出张力。
Hang the same block from a newton meter in water and the reading is less than the weight in air: reading equals weight minus upthrust.
把同一方块挂在水中的弹簧测力计上， 读数小于空气中的重量：读数等于重量减去浮力。
On a planet with smaller g, both weight and upthrust shrink in the same ratio, so a floating object still floats with the same fraction below the surface.
在重力加速度更小的星球上， 重量和浮力按同一比例缩小，所以漂浮物体浸在表面以下的比例不变。
One last distinction that exam marks love.
考试爱考的最后一个区分。
Mass is how much matter you have, in kilograms.
质量是有多少物质，单位千克。
Weight is the gravitational force on that mass, in newtons — mass times g.
重量是质量所受的万有引力，单位牛顿—— 质量乘以重力加速度。
Density is mass per unit volume, in kilograms per cubic metre.
密度是单位体积的质量，单位是每立方米千克。
Do not mix the three, and always give the base unit.
三者不要混，并且每次都写出基本单位。
If a question gives mass and you need a moment, convert to weight first.
若题目给了质量而你需要力矩，先换成重量。
If it asks for density, never answer with a force.
若问的是密度，绝不要用一个力来回答。
Three marks to secure.
三个要拿稳的分。
First, for equilibrium you need both: zero resultant force and zero resultant moment.
第一，平衡需要两者兼备：合力为零，且合力矩为零。
Second, take moments about the pivot where an unknown force acts, so it drops out of the equation.
第二，对未知力所在的支点取力矩，这样它就从方程里消掉。
Third, remember Archimedes: the upthrust equals the weight of the fluid displaced, the density, times gravity, times the volume.
第三，记住阿基米德： 浮力等于所排开流体的重量，也就是密度乘以重力加速度再乘以体积。
Nail these, and this topic is yours.
搞定这些，这个专题就是你的了。
The fixed-wording definitions, one answer only.
固定措辞的定义，只给一个答案。
Centre of gravity: the point at which the whole weight of an object may be considered to act.
重心：可以认为物体全部重力作用于其上的那一点。
Moment of a force: the product of the force and the perpendicular distance from the point to the line of action of the force.
力矩：力与从该点到力的作用线的垂直距离的乘积。
A couple: a pair of equal and opposite forces whose lines of action do not coincide, producing rotation only.
力偶：一对大小相等、方向相反、作用线不重合的力，只产生转动。
Torque of a couple: the product of one of the forces and the perpendicular distance between the lines of action.
力偶矩：其中一个力与两条作用线之间垂直距离的乘积。
Principle of moments: for a body in equilibrium, the sum of the clockwise moments about any point equals the sum of the anticlockwise moments about that point.
力矩原理：对于处于平衡的物体，绕任意一点的顺时针力矩之和等于逆时针力矩之和。
Equilibrium: the resultant force and the resultant torque are both zero.
平衡：合力和合力矩都为零。
Density: mass per unit volume.
密度：单位体积的质量。
Pressure: force per unit area, with the force at right angles to the surface.
压强：单位面积上的力，且该力与表面垂直。
The traps.
陷阱。
Draw the triangle before choosing sine or cosine.
在选正弦还是余弦之前先把三角形画出来。
Upthrust is the weight of the fluid displaced, not the weight of the object.
浮力是被排开流体的重量，不是物体的重量。
Never leave out the weight of a uniform beam — it acts at the centre and has a moment about any pivot that is not there.
绝不要漏掉均匀横梁自身的重力——它作用在中点， 对任何不在中点的支点都有力矩。
Use the perpendicular distance for an angled force, not the distance along the rod.
斜向的力要用垂直距离，不是沿杆量的距离。
And the torque of a couple uses the distance between the two lines of action, so a symmetric couple gives two F d, not F d.
另外力偶矩用的是两条作用线之间的距离， 所以对称的力偶给出的是二倍的 F d，不是 F d。