Circular Motion · 圆周运动
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| uniform circular motion/ˈjuːnɪfɔːm ˈsɜːkjʊlə ˈməʊʃn/ | 匀速圆周运动 | yún sù yuán zhōu yùn dòng |
| centripetal acceleration/senˈtrɪpɪtl əkˌseləˈreɪʃn/ | 向心加速度 | xiàng xīn jiā sù dù |
| centripetal force/senˈtrɪpɪtl fɔːs/ | 向心力 | xiàng xīn lì |
| tangential/tænˈdʒenʃl/ | 切向 | qiè xiàng |
Cut the string and it flies straight
- Whirl a ball on a string in a circle, then suddenly let go.
- It does not curve outward -- it shoots off in a straight line, along the tangent.
- So something was constantly bending its path inward.
- That inward pull is the secret of all circular motion.
剪断绳子,它径直飞出
- 用绳子拴住球在圆周上甩,然后突然松手。
- 它不会向外弯出去——而是沿切线径直射出。
- 所以有某种东西一直在把它的路径向内弯折。
- 那个向内的拉力,正是一切圆周运动的秘密。
Uniform circular motion
- In uniform circular motion 匀速圆周运动 the speed is constant.
- But the direction keeps changing -- so the velocity changes, which means the object is accelerating.
- An object can move at steady speed and still be accelerating, as long as it turns.
匀速圆周运动
- 在匀速圆周运动中速率是恒定的。
- 但方向不断改变——所以速度在变,这意味着物体在加速。
- 只要在转弯,物体就可以以恒定速率运动而仍然在加速。
An object moving in a circle at constant speed is accelerating. · 以恒定速率做圆周运动的物体正在加速。
Its direction changes, so its velocity changes -- that is an acceleration, even at constant speed. · 其方向发生变化，因此速度发生变化——即使速率恒定，这也是一种加速度。
Centripetal acceleration
- The acceleration in a circle is the centripetal acceleration 向心加速度:
- It always points toward the center of the circle.
- Faster motion or a tighter circle (smaller $r$) means a larger inward acceleration.
向心加速度
- 圆周运动中的加速度是向心加速度:
- 它总是指向圆的中心。
- 运动更快或圆更紧(更小的 $r$),意味着更大的向内加速度。
A car rounds a curve of radius $50\ \text{m}$ at $10\ \tfrac{\text{m}}{\text{s}}$. What is its centripetal acceleration (in $\tfrac{\text{m}}{\text{s}^2}$)? · 一辆汽车以速度 $50\ \text{m}$ 通过半径为 $10\ \tfrac{\text{m}}{\text{s}}$的弯道。其向心加速度（单位为 $\tfrac{\text{m}}{\text{s}^2}$)?
$a_c = v^2/r = 100/50 = 2\ \tfrac{\text{m}}{\text{s}^2}$, directed toward the center. · $a_c = v^2/r = 100/50 = 2\ \tfrac{\text{m}}{\text{s}^2}$，方向指向圆心。
The centripetal acceleration points... · 向心加速度指向...
"Centripetal" means center-seeking -- always toward the center of the circle. · “向心”意为朝向中心——始终指向圆的中心。
Centripetal force
- By $\sum F = ma$, that inward acceleration needs an inward centripetal force 向心力:
- This is not a new kind of force -- it is a role filled by tension, gravity, friction, or a normal force.
- Identify which real force points toward the center; that force is the centripetal force.
向心力
- 根据 $\sum F = ma$,那个向内的加速度需要一个向内的向心力:
- 这不是一种新的力——它是一个由张力、重力、摩擦力或法向力充当的角色。
- 找出哪个真实的力指向中心;那个力就是向心力。
What provides the centripetal force? · 什么提供了向心力？
Uniform circular motion needs a net force toward the centre. Match each case to its source. · 匀速圆周运动需要指向圆心的合外力。将每种情况与其来源匹配。
For a car turning on a flat road, what provides the centripetal force? · 对于在平坦路面上转弯的汽车，什么提供了向心力？
Friction points inward and supplies the centripetal force -- which is why a slippery road makes cars skid outward. · 摩擦力向内并提供了向心力——这就是为什么湿滑路面会导致汽车向外打滑的原因。
The centripetal force is not a new force -- it is a ____ filled by a real force like tension or gravity. · 向心力不是一种新力——它是一种 ____，由张力或重力等真实力来充当。
Always identify which real inward force is acting; that force is the centripetal force. · 始终识别哪种真实的向内力在起作用；该力就是向心力。
A $2\ \text{kg}$ ball swings on a string in a circle of radius $1\ \text{m}$ at $3\ \tfrac{\text{m}}{\text{s}}$. What tension is needed (in N)? · 一个$2\ \text{kg}$球在半径为$1\ \text{m}$的圆圈中以$3\ \tfrac{\text{m}}{\text{s}}$摆动。需要的张力是多少（单位为N）？
$F_c = mv^2/r = (2 \times 9)/1 = 18\ \text{N}$, pointing toward the center. · $F_c = mv^2/r = (2 \times 9)/1 = 18\ \text{N}$，指向圆心。
When the speed also changes
- If the object also speeds up or slows down, the force has two parts.
- A tangential 切向 component changes the speed (along the motion).
- The radial (centripetal) component still bends the path inward.
当速率也在变化时
- 如果物体还在加速或减速,力就有两部分。
- 切向分量改变速率(沿运动方向)。
- 径向(向心)分量仍然把路径向内弯折。
A $0.5\ \text{kg}$ ball on a string swings in a circle of radius $2\ \text{m}$ at $4\ \tfrac{\text{m}}{\text{s}}$.
- Centripetal acceleration: $a_c = \dfrac{v^2}{r} = \dfrac{16}{2} = 8\ \tfrac{\text{m}}{\text{s}^2}$.
- Tension needed: $F_c = \dfrac{mv^2}{r} = \dfrac{0.5 \times 16}{2} = 4\ \text{N}$, pointing inward.
一个 $0.5\ \text{kg}$ 的球用绳子拴着,在半径 $2\ \text{m}$ 的圆周上以 $4\ \tfrac{\text{m}}{\text{s}}$ 转动。
- 向心加速度:$a_c = \dfrac{v^2}{r} = \dfrac{16}{2} = 8\ \tfrac{\text{m}}{\text{s}^2}$。
- 所需张力:$F_c = \dfrac{mv^2}{r} = \dfrac{0.5 \times 16}{2} = 4\ \text{N}$,指向内。
There is no outward "centrifugal force" pushing the ball out. The only real force is the inward tension. The outward feeling comes from your body's inertia trying to go straight -- cut the string and the ball proves it by flying off tangent, not outward.
并没有一个把球向外推的"离心力"。唯一真实的力是向内的张力。向外的感觉来自你身体的惯性想要径直前进——剪断绳子,球就以沿切线飞出、而非向外飞出证明了这一点。
In uniform circular motion, speed is constant but the object still accelerates -- the centripetal acceleration $a_c = v^2/r$ points to the center. A real inward force fills the centripetal force role $F_c = mv^2/r$ (tension, gravity, friction...). If the speed also changes, add a tangential component.
在匀速圆周运动中,速率恒定但物体仍在加速——向心加速度 $a_c = v^2/r$ 指向中心。一个真实的向内的力充当向心力角色 $F_c = mv^2/r$(张力、重力、摩擦力……)。如果速率也在变,就加上一个切向分量。