Circular Motion · 圆周运动
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| centripetal/senˈtrɪpɪtl/ | 向心的 | xiàng xīn de |
Whirl a ball on a string — where does it go if it snaps?
- Spin a ball on a string in a circle at steady speed. Now imagine the string snaps.
- It does not fly outward — it shoots off along the tangent, in a straight line.
- So something must constantly pull it inward to bend its path into a circle.
- That inward pull is the key to all circular motion.
用绳甩一个球——绳断了它往哪飞?
- 用绳把球以稳定速率甩成一个圆。现在设想绳断了。
- 它不会向外飞——它沿切线方向,以直线射出。
- 所以一定有某个东西不断地把它向内拉,才把它的路径弯成一个圆。
- 这个向内的拉力是一切圆周运动的关键。
Constant speed, changing velocity
- In uniform circular motion the speed is constant but the velocity is not.
- Velocity is a vector, and its direction changes every instant around the circle.
- A changing velocity means there is an acceleration — even at constant speed.
- This acceleration points toward the centre, so we call it centripetal 向心的.
速率恒定,速度却在变
- 在匀速圆周运动中,速率恒定,但速度不恒定。
- 速度是矢量,它的方向在圆上每一瞬间都在改变。
- 速度在变就意味着存在加速度——即使速率恒定。
- 这个加速度指向圆心,所以我们称它为向心的。
A ball whirls on a string in a horizontal circle. The string suddenly snaps. Which way does the ball fly? · 一个球用绳子在水平面内做圆周运动。绳子突然断裂。球飞向哪个方向？
With no inward force, inertia carries the ball in a straight line — along the tangent · 相切 where the string broke. · 没有向内的力，惯性使球沿直线运动——即绳子断裂处的切线方向。
An object in uniform circular motion has zero acceleration because its speed is constant. · 匀速圆周运动的物体加速度为零，因为其速率恒定。
Its direction changes, so its velocity changes — there is a centripetal acceleration $v^2/r$ toward the centre. · 其方向发生变化，因此速度变化——存在指向中心的向心加速度$v^2/r$a_c=v²/r。
Centripetal acceleration and force
- The centripetal acceleration has size $a_c = \dfrac{v^2}{r}$, directed to the centre.
- By Newton's second law, a net centripetal force $F_c = \dfrac{mv^2}{r}$ must point inward.
- Faster spin or a tighter circle both demand a larger inward force.
- This force is not new — it is provided by tension, gravity, friction, or a normal force.
向心加速度与向心力
- 向心加速度大小为 $a_c = \dfrac{v^2}{r}$,指向圆心。
- 由牛顿第二定律,一个合的向心力 $F_c = \dfrac{mv^2}{r}$ 必须指向内。
- 转得更快或圆更小,都需要更大的向内的力。
- 这个力不是新的力——它由张力、重力、摩擦力或法向力提供。
Around the circle · 绕圆周运动
Sweep the angle and change the radius to see how position and direction change around a circle. · 扫过角度并改变半径，观察位置和方向如何在圆周上变化。
A $2\ \text{kg}$ ball moves at $4\ \tfrac{\text{m}}{\text{s}}$ in a circle of radius $8\ \text{m}$. What centripetal force is needed, in $\text{N}$? · 一个质量为 $2\ \text{kg}$ 的球以 $4\ \tfrac{\text{m}}{\text{s}}$ 的速度在半径为 $8\ \text{m}$ 的圆中运动。所需的向心力是多少，单位为 $\text{N}$？
$F_c = \dfrac{mv^2}{r} = \dfrac{2 \times 16}{8} = 4\ \text{N}$, toward the centre. · $F_c = \dfrac{mv^2}{r} = \dfrac{2 \times 16}{8} = 4\ \text{N}$F_c=mv²/r，指向中心。
In which direction does the net (centripetal) force point? · 合力（向心力）指向哪个方向？
The net force is centripetal — it points toward the centre, bending the path into a circle. · 合力是向心力——它指向中心，使路径弯曲成圆形。
Centripetal acceleration has magnitude $v^2 / \_\_$ (fill in the missing symbol). · 向心加速度的大小为$v^2 / \_\_$v²/r（填入缺失符号）。
$a_c = v^2/r$: bigger speed or smaller radius means a larger centripetal acceleration. · $a_c = v^2/r$a_c=v²/r：更大的速度或更小的半径意味着更大的向心加速度。
Select all · 所有 forces that can provide the centripetal force in real situations. · 选择所有能在实际情境中提供向心力的力。
Tension, gravity and friction can each supply the inward force. There is no real outward centrifugal force. · 张力、重力和摩擦力均可提供向内的力。不存在真实的向外离心力。
There is no outward "centrifugal" force
- The feeling of being flung outward in a turning car is your inertia, not a real force.
- Your body "wants" to go straight (tangent); the car door pushes you inward to turn you.
- The only real force on you is that inward push — there is no outward force acting.
- Call the inward net force centripetal; the outward "centrifugal force" is a fiction.
不存在向外的"离心力"
- 转弯的车里那种被甩向外的感觉,是你的惯性,而不是真实的力。
- 你的身体"想"沿直线(切线)走;车门把你向内推,才让你转弯。
- 作用在你身上的唯一真实的力就是那个向内的推——没有向外的力在作用。
- 把向内的合力叫作向心力;向外的"离心力"是虚构的。
If the string breaks, the ball flies off along the tangent, not radially outward. And there is no real outward "centrifugal force" — the outward feeling is just inertia resisting the inward turn. The genuine net force is inward.
如果绳断了,球沿切线飞出,而不是沿半径向外。而且不存在真实的向外"离心力"——向外的感觉只是惯性在抵抗向内的转弯。真正的合力是向内的。
A $2\ \text{kg}$ ball moves at $4\ \tfrac{\text{m}}{\text{s}}$ in a circle of radius $8\ \text{m}$.
- $F_c = \dfrac{mv^2}{r} = \dfrac{2 \times 4^2}{8} = \dfrac{32}{8} = 4\ \text{N}$, directed toward the centre.
The string tension must supply this $4\ \text{N}$ inward.
一个 $2\ \text{kg}$ 的球在半径 $8\ \text{m}$ 的圆上以 $4\ \tfrac{\text{m}}{\text{s}}$ 运动。
- $F_c = \dfrac{mv^2}{r} = \dfrac{2 \times 4^2}{8} = \dfrac{32}{8} = 4\ \text{N}$,指向圆心。
绳的张力必须提供这向内的 $4\ \text{N}$。
Uniform circular motion has constant speed but changing velocity, so there is a centripetal acceleration $a_c = v^2/r$ toward the centre. A net centripetal force $F_c = mv^2/r$ (from tension, gravity, friction...) points inward. There is no real outward "centrifugal" force — that is inertia.
匀速圆周运动速率恒定但速度在变,所以存在指向圆心的向心加速度 $a_c = v^2/r$。合的向心力 $F_c = mv^2/r$(由张力、重力、摩擦力……提供)指向内。不存在真实的向外"离心"力——那是惯性。