Rotational Inertia · 转动惯量
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| rotational inertia/rəʊˈteɪʃənl ɪˈnɜːʃə/ | 转动惯量 | zhuǎn dòng guàn liàng |
A skater spins faster by pulling her arms in
- A figure skater spins slowly with arms outstretched, then pulls them in and whirls into a blur.
- She added no push — she just moved her mass closer to the axis.
- How hard something is to spin depends on where its mass sits, not only how much.
- This "rotational stubbornness" is called rotational inertia 转动惯量.
滑冰运动员收臂就转得更快
- 花样滑冰运动员张开双臂慢慢旋转,然后把手臂收拢,转成一团模糊。
- 她没有增加任何推力——她只是把质量移得更靠近轴。
- 一样东西有多难旋转,取决于它的质量在哪里,而不只是有多少。
- 这种"转动上的固执"叫做转动惯量。
Mass distribution is everything
- Rotational inertia (moment of inertia) $I$ measures resistance to angular acceleration.
- For point masses: $I = \sum m r^2$ — each bit of mass counts by its distance squared.
- Mass far from the axis contributes far more than mass near it.
- Units are $\text{kg}\cdot\text{m}^2$.
质量分布决定一切
- 转动惯量 $I$ 量度对角加速度的抵抗。
- 对质点:$I = \sum m r^2$——每一小块质量都按它距离的平方计入。
- 离轴远的质量,比离轴近的贡献大得多。
- 单位是 $\text{kg}\cdot\text{m}^2$。
Two $0.5\ \text{kg}$ balls sit $2\ \text{m}$ from the axis on a light rod. What is the rotational inertia, in $\text{kg}\cdot\text{m}^2$? · 两个$0.5\ \text{kg}$的小球位于距轴$2\ \text{m}$处的一根轻杆上。转动惯量是多少，单位为$\text{kg}\cdot\text{m}^2$？
$I = \sum mr^2 = 2 \times (0.5 \times 2^2) = 4\ \text{kg}\cdot\text{m}^2$.
Rotational inertia depends on: · 转动惯量取决于：
$I = \sum mr^2$ depends on both the mass and its distance from the axis (squared). · $I = \sum mr^2$取决于质量和其距轴的距离（平方）。
For point masses, rotational inertia is the sum of $m r^{\_\_}$. Fill in the power. · 对于质点，转动惯量是 $m r^{\_\_}$ 的总和。填入幂次。
$I = \sum m r^2$ — distance appears squared · 平方. · $I = \sum m r^2$ ——距离出现为平方。
Move the same two $0.5\ \text{kg}$ balls to $1\ \text{m}$ from the axis. What is the new rotational inertia, in $\text{kg}\cdot\text{m}^2$? · 将同样的两个 $0.5\ \text{kg}$ 球移动到距轴 $1\ \text{m}$ 的位置。新的转动惯量是多少，单位为 $\text{kg}\cdot\text{m}^2$？
$I = 2 \times (0.5 \times 1^2) = 1\ \text{kg}\cdot\text{m}^2$ — halving $r$ quarters $I$. · $I = 2 \times (0.5 \times 1^2) = 1\ \text{kg}\cdot\text{m}^2$ — 将$r$减半会使$I$变为四分之一。
Shape matters, so it has standard formulas
- The same mass can have very different $I$ depending on its shape and axis.
- A hollow ring (all mass at the rim) has more $I$ than a solid disk of the same mass.
- Standard shapes have known formulas, e.g. a solid disk is $I = \tfrac12 M R^2$.
- The $r^2$ weighting is why hollow objects feel "heavier" to spin.
形状很重要,所以有标准公式
- 同样的质量,依形状和轴的不同,可以有截然不同的 $I$。
- 空心圆环(质量全在边缘)比同质量的实心圆盘 $I$ 更大。
- 标准形状有已知公式,例如实心圆盘是 $I = \tfrac12 M R^2$。
- 正是 $r^2$ 加权,使空心物体旋转起来"感觉更重"。
Two objects have the same mass. Select all · 所有 ways to make one harder to spin than the other. · 两个物体质量相同。选择所有使其中一个比另一个更难旋转的方法。
Moving mass outward (rim, hollow ring) raises $I$. Colour has no effect. · 向外移动质量（边缘、空心环）会增加$I$。颜色没有影响。
Back to the skater
- Pulling her arms in shrinks $r$, so her $I$ drops sharply (because of the $r^2$).
- With angular momentum conserved (next topic), a smaller $I$ means a larger $\omega$ — she speeds up.
- Divers and gymnasts tuck for the same reason: small $I$, fast spin.
- Extend again and the spin slows back down.
回到滑冰者
- 收臂使 $r$ 变小,所以她的 $I$ 急剧下降(因为 $r^2$)。
- 在角动量守恒下(下一主题),更小的 $I$ 意味着更大的 $\omega$——她加快了。
- 跳水和体操运动员团身也是同样的道理:小 $I$、快旋转。
- 再次伸展,旋转又慢下来。
More or less rotational inertia? · 转动惯量更大还是更小？
Rotational inertia depends on how far the mass sits from the axis. Sort each case. · 转动惯量取决于质量距离轴的远近。对每种情况进行排序。
A spinning skater speeds up when she pulls her arms in because her rotational inertia decreases. · 旋转的花样滑冰运动员收拢手臂时会加速，因为她的转动惯量减小了。
Smaller $r$ means smaller $I$; with angular momentum conserved, $\omega$ rises. · 较小的$r$意味着较小的$I$；在角动量守恒的情况下，$\omega$会上升。
Rotational inertia is not just mass — it depends on how the mass is arranged. Two objects of equal mass can be very different to spin. Because of the $r^2$, mass at the rim matters far more than mass near the axis.
转动惯量不只是质量——它取决于质量如何排布。两个质量相等的物体旋转起来可以大不相同。因为 $r^2$,边缘处的质量比靠近轴的质量重要得多。
Two $0.5\ \text{kg}$ balls sit on a light rod, each $2\ \text{m}$ from the axis.
- $I = \sum m r^2 = 2 \times (0.5 \times 2^2) = 2 \times 2 = 4\ \text{kg}\cdot\text{m}^2$.
Move them to $1\ \text{m}$ and $I$ drops to $2 \times (0.5 \times 1) = 1\ \text{kg}\cdot\text{m}^2$ — four times smaller.
两个 $0.5\ \text{kg}$ 的球位于一根轻杆上,各距轴 $2\ \text{m}$。
- $I = \sum m r^2 = 2 \times (0.5 \times 2^2) = 2 \times 2 = 4\ \text{kg}\cdot\text{m}^2$。
把它们移到 $1\ \text{m}$,$I$ 降到 $2 \times (0.5 \times 1) = 1\ \text{kg}\cdot\text{m}^2$——小了四倍。
Rotational inertia $I$ is the resistance to angular acceleration: $I = \sum mr^2$. It depends on where the mass sits (distance squared), not just how much — so a skater pulling her arms in lowers $I$ and spins faster. Units: $\text{kg}\cdot\text{m}^2$.
转动惯量 $I$ 是对角加速度的抵抗:$I = \sum mr^2$。它取决于质量在哪里(距离的平方),而不只是有多少——所以滑冰者收臂降低 $I$、转得更快。单位:$\text{kg}\cdot\text{m}^2$。