Connecting Linear and Rotational Motion · 连接直线运动和转动运动
On a merry-go-round, the edge whips past
- Stand near the centre of a spinning merry-go-round and you drift slowly.
- Stand at the edge and you are whipped around fast — yet everyone shares the same spin.
- The whole platform turns together, so everyone has the same angular velocity.
- But each rider's linear speed depends on how far out they sit.
在旋转木马上,边缘飞速掠过
- 站在旋转木马的中心附近,你只是慢慢地漂移。
- 站在边缘,你被飞快地甩着转——可每个人共享同一个旋转。
- 整个平台一起转,所以每个人都有相同的角速度。
- 但每位乘客的线速度取决于他坐得多靠外。
The bridge equations
- Arc length links to angle: $s = r\theta$.
- Linear speed links to angular speed: $v = r\omega$.
- Tangential acceleration links to angular acceleration: $a_t = r\alpha$.
- In every case, multiply the angular quantity by the radius $r$.
桥梁方程
- 弧长与角度相联系:$s = r\theta$。
- 线速率与角速率相联系:$v = r\omega$。
- 切向加速度与角加速度相联系:$a_t = r\alpha$。
- 每一种情形,都是把角量乘以半径 $r$。
Radius sets the speed · 半径决定线速度
Change the radius and see how the arc length (and so the linear speed) scales with r. · 改变半径，观察弧长（以及线速度）如何随r缩放。
A point is $0.3\ \text{m}$ from a wheel's axis, spinning at $\omega = 10\ \tfrac{\text{rad}}{\text{s}}$. What is its linear speed, in $\tfrac{\text{m}}{\text{s}}$? · 一点距离轮轴$0.3\ \text{m}$，轮子转速为$\omega = 10\ \tfrac{\text{rad}}{\text{s}}$。该点的线速度是多少，单位为$\tfrac{\text{m}}{\text{s}}$？
$v = r\omega = 0.3 \times 10 = 3\ \tfrac{\text{m}}{\text{s}}$.
The arc length swept is $s = r\_\_$ (fill in the angular quantity). · 扫过的弧长为 $s = r\_\_$（填入角量）。
$s = r\theta$ — radius times the angle in radians. · $s = r\theta$ — 半径乘以弧度单位下的角度。
Select all · 所有 correct linear–rotational bridge equations. · 选择所有正确的直线-转动桥梁方程。
Each links an angular quantity to a linear one by multiplying by $r$. $v = \omega/r$ is wrong. · 每个方程通过将$r$乘以一个转动量来链接一个转动量和一个直线量。$v = \omega/r$是错误的。
Farther out means faster
- Because $v = r\omega$, a larger radius gives a larger linear speed for the same spin.
- The rim of a wheel moves faster than a point near the hub.
- This is why a long lever tip, a fan blade edge, or the outer horse on a carousel moves quickest.
- Same $\omega$ everywhere on the object; different $v$ at different radii.
越靠外越快
- 因为 $v = r\omega$,同样的旋转下,更大的半径给出更大的线速率。
- 轮子的外缘比靠近轮毂的点动得快。
- 这就是为什么长杠杆的末端、风扇叶片的边缘,或旋转木马最外圈的马动得最快。
- 物体上处处 $\omega$ 相同;不同半径处 $v$ 不同。
On a rigid spinning disk, how do a rim point and a point near the hub compare? · 在刚性旋转圆盘上，边缘点和靠近轴心的点相比如何？
All points share $\omega$; the rim has a larger radius, so $v = r\omega$ is larger there. · 所有点共享$\omega$；边缘半径更大，因此那里的$v = r\omega$也更大。
For the same wheel above, what is the linear speed at $0.6\ \text{m}$ from the axis, in $\tfrac{\text{m}}{\text{s}}$? · 对于上述同一个轮子，距离轴心$0.6\ \text{m}$处的线速度是多少，单位为$\tfrac{\text{m}}{\text{s}}$？
$v = r\omega = 0.6 \times 10 = 6\ \tfrac{\text{m}}{\text{s}}$ — double the radius, double the speed. · $v = r\omega = 0.6 \times 10 = 6\ \tfrac{\text{m}}{\text{s}}$ — 半径加倍，速度加倍。
Rolling without slipping
- When a wheel rolls without slipping, its centre moves at $v = r\omega$.
- The contact point is momentarily at rest; the top moves at $2v$.
- This neat condition ties a wheel's spinning to its travelling.
- We will use it again for rolling energy in the next topic.
无滑滚动
- 当轮子无滑动地滚动时,它的中心以 $v = r\omega$ 运动。
- 接触点在那一瞬间静止;顶部以 $2v$ 运动。
- 这个简洁的条件把轮子的旋转与它的行进联系起来。
- 我们会在下一主题的滚动能量中再次用到它。
A wheel rolling without slipping has its centre moving at $v = r\omega$. · 无滑动滚动的轮子，其质心移动速度为$v = r\omega$。
Rolling without slipping ties travel to spin exactly by $v = r\omega$. · 无滑动滚动通过$v = r\omega$将平移与旋转精确关联起来。
Every point on a rigid spinning object shares the same $\omega$, but not the same $v$. Do not assume the edge and the centre move at the same linear speed — the edge is much faster because its radius is larger ($v = r\omega$).
刚性旋转物体上的每一点共享相同的 $\omega$,但不共享相同的 $v$。不要以为边缘和中心以相同的线速率运动——边缘快得多,因为它的半径更大($v = r\omega$)。
A point sits $0.3\ \text{m}$ from the axis of a wheel turning at $\omega = 10\ \tfrac{\text{rad}}{\text{s}}$.
- $v = r\omega = 0.3 \times 10 = 3\ \tfrac{\text{m}}{\text{s}}$.
A point at $0.6\ \text{m}$ would move at $6\ \tfrac{\text{m}}{\text{s}}$ — twice as fast, for the same spin.
一个点位于以 $\omega = 10\ \tfrac{\text{rad}}{\text{s}}$ 旋转的轮子轴线外 $0.3\ \text{m}$ 处。
- $v = r\omega = 0.3 \times 10 = 3\ \tfrac{\text{m}}{\text{s}}$。
$0.6\ \text{m}$ 处的点会以 $6\ \tfrac{\text{m}}{\text{s}}$ 运动——同样的旋转下快一倍。
Linear and angular motion are linked by the radius: $s = r\theta$, $v = r\omega$, $a_t = r\alpha$. Every point on a rigid body shares the same $\omega$, but points farther from the axis have larger $v$. A wheel rolling without slipping obeys $v = r\omega$.
线运动与角运动由半径联系:$s = r\theta$、$v = r\omega$、$a_t = r\alpha$。刚体上每一点 $\omega$ 相同,但离轴越远的点 $v$ 越大。无滑滚动的轮子遵守 $v = r\omega$。