Rotational Kinetic Energy · 转动动能
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| rotational kinetic energy/rəʊˈteɪʃənl kɪˈnetɪk ˈenədʒi/ | 转动动能 | zhuǎn dòng dòng néng |
| angular speed/ˈæŋɡjʊlə spiːd/ | 角速度 | jiǎo sù dù |
The wheel that stores its spin
- A heavy potter's wheel keeps turning long after you let go.
- A spinning flywheel can drive a machine for seconds on its own.
- All that motion is stored, ready to do work.
- A spinning object carries a store of motion we can calculate.
储存旋转的轮子
- 沉重的陶轮在你松手后仍会转很久。
- 旋转的飞轮可以独自驱动机器好几秒。
- 那些运动都被储存起来,随时可以做功。
- 旋转的物体携带着一份可以计算的运动储备。
Kinetic energy of a spin
- A spinning object has rotational kinetic energy 转动动能:
- $I$ is the rotational inertia, $\omega$ the angular speed 角速度.
- It mirrors $\tfrac{1}{2}mv^2$, with $I$ for mass and $\omega$ for speed.
旋转的动能
- 旋转的物体具有转动动能:
- $I$ 是转动惯量,$\omega$ 是角速度。
- 它与 $\tfrac{1}{2}mv^2$ 对应,$I$ 代替质量,$\omega$ 代替速度。
A wheel with $I = 2\ \text{kg}\cdot\text{m}^2$ spins at $\omega = 3\ \text{rad/s}$. Its rotational kinetic energy (in J)? · 一个转动惯量为$I = 2\ \text{kg}\cdot\text{m}^2$的轮子以角速度$\omega = 3\ \text{rad/s}$旋转。其转动动能（单位 J）是多少？
$K = \tfrac{1}{2}I\omega^2 = \tfrac{1}{2}(2)(3)^2 = 9\ \text{J}$.
Rotational kinetic energy is one-half times I times omega ____. · 转动动能等于二分之一乘以 I 乘以 omega ____。
$K = \tfrac{1}{2}I\omega^2$ -- omega is squared. · $K = \tfrac{1}{2}I\omega^2$——omega 是被平方的。
Why the square matters
- Double the angular speed and you get four times the energy.
- $\omega$ is squared, just like $v$ in $\tfrac{1}{2}mv^2$.
- Doubling $I$ only doubles the energy -- speed is the stronger lever.
平方为何重要
- 角速度加倍,能量变成四倍。
- $\omega$ 是平方的,正如 $\tfrac{1}{2}mv^2$ 里的 $v$。
- $I$ 加倍只让能量翻倍——速度是更强的杠杆。
If you triple a wheel's angular speed but keep its rotational inertia, its rotational kinetic energy becomes... · 如果你将轮子的角速度增至三倍但保持其转动惯量不变，其转动动能变为...
$\omega$ is squared, so $3^2 = 9$ times the energy. · $\omega$被平方，所以$3^2 = 9$倍的能量。
Rolling: two motions at once
- A rolling wheel moves forward and spins.
- Its total kinetic energy adds both parts:
- The centre-of-mass motion, plus the spin about that centre.
滚动:两种运动同时进行
- 滚动的轮子既向前移动又自转。
- 它的总动能把两部分相加:
- 质心的平动,加上绕该质心的自转。
Rotational kinetic energy · 转动动能
A spinning body's kinetic energy grows with the square of its angular speed. · 旋转体的动能随角速度的平方增长。
A rolling ball's total kinetic energy includes both a translational and a rotational term. · 滚动球的总动能包括平动项和转动项。
$K = \tfrac{1}{2}mv^2 + \tfrac{1}{2}I\omega^2$ -- it moves and spins. · $K = \tfrac{1}{2}mv^2 + \tfrac{1}{2}I\omega^2$——它既移动又旋转。
A rolling object has $\tfrac{1}{2}mv^2 = 6\ \text{J}$ and $\tfrac{1}{2}I\omega^2 = 4\ \text{J}$. Its total kinetic energy (in J)? · 滚动物体具有$\tfrac{1}{2}mv^2 = 6\ \text{J}$和$\tfrac{1}{2}I\omega^2 = 4\ \text{J}$。其总动能（单位 J）是多少？
Add the two shares, $6 + 4 = 10\ \text{J}$. · 将两部分相加，$6 + 4 = 10\ \text{J}$。
A solid disc with $I = 0.5\ \text{kg}\cdot\text{m}^2$ spins at $\omega = 4\ \text{rad/s}$.
- $K_{rot} = \tfrac{1}{2}I\omega^2 = \tfrac{1}{2}(0.5)(4)^2 = 4\ \text{J}$.
- Speed it up to $8\ \text{rad/s}$ and it stores $16\ \text{J}$ -- four times as much.
一个实心圆盘 $I = 0.5\ \text{kg}\cdot\text{m}^2$,以 $\omega = 4\ \text{rad/s}$ 旋转。
- $K_{rot} = \tfrac{1}{2}I\omega^2 = \tfrac{1}{2}(0.5)(4)^2 = 4\ \text{J}$。
- 加速到 $8\ \text{rad/s}$,它储存 $16\ \text{J}$——四倍之多。
Before using $K = \tfrac{1}{2}I\omega^2$, the angular speed $\omega$ must be in... · 在使用$K = \tfrac{1}{2}I\omega^2$之前，角速度$\omega$必须是...
SI rotational formulas need $\omega$ in rad/s; convert rpm or rev/s first. · SI 转动公式需要$\omega$单位为 rad/s；需先将 rpm 或 rev/s 转换。
Two traps. First, $\omega$ must be in radians per second, not rev/s or rpm. Second, anything that rolls has both terms: a rolling ball carries $\tfrac{1}{2}mv^2$ and $\tfrac{1}{2}I\omega^2$. Leaving out the spin term undercounts the energy.
两个陷阱。第一,$\omega$ 必须用弧度每秒,而不是转每秒或转每分。第二,任何滚动的东西都有两项:滚动的球既有 $\tfrac{1}{2}mv^2$ 又有 $\tfrac{1}{2}I\omega^2$。漏掉自转项会低估能量。
A spinning object stores rotational kinetic energy $K_{rot} = \tfrac{1}{2}I\omega^2$ -- the twin of $\tfrac{1}{2}mv^2$, with $I$ for mass and angular speed for speed. Because $\omega$ is squared, doubling the spin quadruples the energy. A rolling object owns both a translational and a rotational share.
旋转的物体储存转动动能 $K_{rot} = \tfrac{1}{2}I\omega^2$——$\tfrac{1}{2}mv^2$ 的孪生,$I$ 代替质量、角速度代替速度。因为 $\omega$ 是平方的,旋转加倍会使能量变成四倍。滚动的物体同时拥有平动和转动两份。