Rolling · 滚动
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| rolling without slipping/ˈrəʊlɪŋ wɪˈðaʊt ˈslɪpɪŋ/ | 无滑滚动 | wú huá gǔn dòng |
Why a ball beats a ring downhill
- Race a solid ball and a hollow ring down the same ramp.
- Same mass, same radius, same start -- yet the ball wins every time.
- The secret is how each one shares its motion.
- A wheel that turns as it moves obeys a neat speed rule.
为什么下坡时球比环快
- 让实心球和空心环从同一斜坡上比赛滚下。
- 相同质量、相同半径、相同起点——可球每次都赢。
- 秘密在于各自如何分配它的运动。
- 边转边动的轮子遵守一条简洁的速度规则。
Rolling without slipping
- When a wheel rolls without slipping, its speed and spin lock together:
- The contact point is momentarily at rest on the ground.
- Speeding up obeys the matching rule $a = \alpha r$.
无滑滚动
- 当轮子无滑滚动时,它的速度和自转锁在一起:
- 接触点在地面上瞬时静止。
- 加速时遵守相应的规则 $a = \alpha r$。
A wheel of radius $0.3\ \text{m}$ rolls without slipping at $\omega = 10\ \text{rad/s}$. Its forward speed (in m/s)? · 半径为$0.3\ \text{m}$的轮子以速度$\omega = 10\ \text{rad/s}$纯滚动。其前进速度（单位 m/s）是多少？
$v = \omega r = 10 \times 0.3 = 3\ \text{m/s}$.
For a wheel rolling without slipping, the point touching the ground is momentarily... · 对于纯滚动的轮子，接触地面的点瞬时...
The contact point has zero velocity at that instant -- that is the no-slip condition. · 接触点在该瞬间的速度为零——这就是无滑动条件。
While a wheel of radius r rolls without slipping, its linear acceleration a equals alpha times . · 当半径为r的轮子无滑动滚动时，其线加速度a等于alpha乘以。
$a = \alpha r$ -- the acceleration version of $v = \omega r$. · $a = \alpha r$ -- $v = \omega r$的加速度版本。
Motion shared two ways
- A rolling object's kinetic energy splits into two parts:
- Part goes into moving forward, part into spinning.
- Using $v = \omega r$, both parts grow together.
运动分成两份
- 滚动物体的动能分成两部分:
- 一部分用于向前移动,一部分用于自转。
- 用 $v = \omega r$,两部分一起增大。
A rolling object's kinetic energy is split between translation and rotation. · 滚动物体的动能分为平动和转动两部分。
$K = \tfrac{1}{2}mv^2 + \tfrac{1}{2}I\omega^2$ -- moving and spinning. · $K = \tfrac{1}{2}mv^2 + \tfrac{1}{2}I\omega^2$ -- 既在移动又在旋转。
Who reaches the bottom first
- On a ramp, the released energy splits between forward motion and spin.
- A shape with less rotational inertia (per $mr^2$) keeps more for speed.
- Order: solid sphere, then solid cylinder, then hollow ring -- the sphere wins.
谁先到达底部
- 在斜坡上,释放的能量在向前运动与自转之间分配。
- 转动惯量更小(按 $mr^2$ 计)的形状,把更多能量留给速度。
- 顺序:实心球,然后实心圆柱,再是空心环——球获胜。
Rolling without slipping · 纯滚动（无滑动滚动）
For a wheel that rolls without slipping, v = r omega. Sort each case. · 对于纯滚动的轮子，v = r omega。对每种情况进行排序。
Order these from FIRST to LAST reaching the bottom of the same ramp (same m, r). · 按从最先到最晚到达同一斜面底部的顺序排列这些物体（质量m、半径r相同）。
Smaller rotational inertia keeps more energy for speed, so the sphere leads and the ring trails. · 较小的转动惯量使更多能量用于速度，因此球体领先，圆环落后。
A hoop and a solid disc, same $m$ and $r$, roll from rest down the same ramp.
- Hoop: $I = mr^2$, so half its energy goes to spin -- it is slow.
- Disc: $I = \tfrac{1}{2}mr^2$, a smaller spin share -- it arrives first.
一个圆环和一个实心圆盘,$m$ 和 $r$ 相同,从静止沿同一斜坡滚下。
- 圆环:$I = mr^2$,所以一半能量给了自转——它慢。
- 圆盘:$I = \tfrac{1}{2}mr^2$,自转占比更小——它先到。
During rolling without slipping, how much work does the static friction do? · 在无滑动滚动过程中，静摩擦力做了多少功？
No sliding at the contact means no friction work and no heat -- unlike a skidding wheel. · 接触点无滑动意味着没有摩擦做功，也没有热量产生——这与打滑的轮子不同。
Rolling without slipping needs static friction at the contact, but that friction does no work -- the contact point does not slide, so there is no friction-heat loss. Only a skidding (slipping) wheel loses energy to kinetic friction. And $v = \omega r$ links speed to spin only while it rolls without slipping.
无滑滚动需要接触处的静摩擦,但那摩擦不做功——接触点不滑动,所以没有摩擦生热的损失。只有打滑(滑动)的轮子才把能量损失给动摩擦。而且 $v = \omega r$ 只在无滑滚动时把速度和自转联系起来。
Rolling without slipping 无滑滚动 locks speed to spin: $v = \omega r$ (and $a = \alpha r$). The kinetic energy splits into $\tfrac{1}{2}mv^2 + \tfrac{1}{2}I\omega^2$, so a shape with smaller rotational inertia wins a downhill race (sphere before cylinder before ring). Static friction enables the roll but does no work.
无滑滚动把速度锁到自转:$v = \omega r$(以及 $a = \alpha r$)。动能分成 $\tfrac{1}{2}mv^2 + \tfrac{1}{2}I\omega^2$,所以转动惯量更小的形状会赢得下坡比赛(球先于圆柱先于环)。静摩擦使滚动成为可能但不做功。