Frequency and Period of SHM · SHM的频率与周期
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| period/ˈpɪərɪəd/ | 周期 | zhōu qī |
| frequency/ˈfriːkwənsi/ | 频率 | pín lǜ |
| angular frequency/ˈæŋɡjʊlə ˈfriːkwənsi/ | 角频率 | jiǎo pín lǜ |
Counting the beats
- A metronome ticks at a steady rate you can count.
- A heavy mass on a soft spring bobs slowly; a stiff spring snaps back quickly.
- Each oscillator has its own natural rhythm.
- Two simple formulas predict exactly how fast each one repeats.
数节拍
- 节拍器以你能数出的稳定速率滴答作响。
- 软弹簧上的重物块慢慢起伏;硬弹簧则迅速弹回。
- 每个振子都有自己天然的节奏。
- 两条简单的公式能精确预测各自重复得多快。
Period, frequency, and omega
- The period 周期 $T$ is the time for one full cycle.
- The frequency 频率 $f = 1/T$ counts cycles per second.
- The angular frequency 角频率 is $\omega = 2\pi f = \dfrac{2\pi}{T}$.
周期、频率与 omega
- 周期 $T$ 是完成一个完整循环的时间。
- 频率 $f = 1/T$ 数每秒的循环数。
- 角频率是 $\omega = 2\pi f = \dfrac{2\pi}{T}$。
An oscillator has a period of $0.25\ \text{s}$. Its frequency (in Hz)? · 某振荡器的周期为$0.25\ \text{s}$。其频率（单位：Hz）是多少？
$f = 1/T = 1/0.25 = 4\ \text{Hz}$.
A mass on a spring
- A mass $m$ on a spring of stiffness $k$ has period:
- A heavier mass swings slower (longer $T$); a stiffer spring is faster.
- It follows from $\omega = \sqrt{k/m}$.
弹簧上的物块
- 质量 $m$ 挂在劲度为 $k$ 的弹簧上,周期为:
- 质量越大摆动越慢($T$ 越长);弹簧越硬越快。
- 它来自 $\omega = \sqrt{k/m}$。
You replace a mass-spring oscillator's mass with a heavier one. The period... · 你将弹簧振子的质量替换为更重的质量。周期...
$T = 2\pi\sqrt{m/k}$, so a bigger $m$ gives a longer $T$. · $T = 2\pi\sqrt{m/k}$，因此更大的$m$导致更长的$T$。
If you quadruple the mass on a spring, the period becomes... · 如果你将弹簧上的质量变为原来的四倍，周期变为...
$T \propto \sqrt{m}$, so $\sqrt{4} = 2$ times the period. · $T \propto \sqrt{m}$，因此$\sqrt{4} = 2$倍的周期。
A swinging pendulum
- A pendulum of length $L$ has period:
- A longer pendulum swings slower; stronger gravity is faster.
- Notice the mass of the bob does not appear.
摆动的单摆
- 长度为 $L$ 的单摆,周期为:
- 单摆越长摆得越慢;引力越强越快。
- 注意摆锤的质量不出现。
What changes the period? · 什么改变了周期？
Sort each change by whether it changes an oscillator's period. · 按每种变化是否改变振荡器周期进行分类。
Making a pendulum's bob heavier (same length) changes its period. · 增加单摆摆锤的质量（长度不变）会改变其周期。
$T = 2\pi\sqrt{L/g}$ has no mass term -- the bob's mass does not matter. · $T = 2\pi\sqrt{L/g}$中没有质量项——摆锤的质量无关紧要。
A pendulum has $L = 1.0\ \text{m}$ and $g = 9.8\ \text{m/s}^2$. Its period (in s, to 1 decimal)? · 单摆具有$L = 1.0\ \text{m}$和$g = 9.8\ \text{m/s}^2$。其周期（单位：s，保留1位小数）是多少？
$T = 2\pi\sqrt{L/g} = 2\pi\sqrt{1/9.8} \approx 2.0\ \text{s}$.
A $0.2\ \text{kg}$ mass on a $k = 50\ \text{N/m}$ spring.
- Period: $T = 2\pi\sqrt{m/k} = 2\pi\sqrt{0.2/50} = 2\pi\sqrt{0.004} \approx 0.40\ \text{s}$.
- Frequency: $f = 1/T \approx 2.5\ \text{Hz}$.
一个 $0.2\ \text{kg}$ 的物块挂在 $k = 50\ \text{N/m}$ 的弹簧上。
- 周期:$T = 2\pi\sqrt{m/k} = 2\pi\sqrt{0.2/50} = 2\pi\sqrt{0.004} \approx 0.40\ \text{s}$。
- 频率:$f = 1/T \approx 2.5\ \text{Hz}$。
For ideal SHM, a larger amplitude gives a longer period. · 对于理想简谐运动，较大的振幅会导致较长的周期。
The period is independent of amplitude -- big and small swings take the same time. · 周期与振幅无关——大摆动和小摆动所需时间相同。
A remarkable fact -- for ideal SHM the period does not depend on amplitude, so a big swing and a small swing take the same time. The pendulum formula holds only for small angles; the spring period ignores gravity and the pendulum period ignores the bob's mass. And mind the square root: quadrupling the mass only doubles the period.
一个了不起的事实——对理想简谐运动,周期不依赖振幅,所以大摆动和小摆动花的时间相同。单摆公式只在小角度成立;弹簧周期忽略引力,单摆周期忽略摆锤质量。还要注意平方根:把质量变四倍只让周期翻倍。
The period $T$ (one cycle), frequency $f = 1/T$, and angular frequency $\omega = 2\pi/T$ describe the rhythm. A spring gives $T = 2\pi\sqrt{m/k}$; a pendulum gives $T = 2\pi\sqrt{L/g}$. The period is independent of amplitude, and each formula ignores one thing -- the spring ignores gravity, the pendulum ignores the bob's mass.
周期 $T$(一个循环)、频率 $f = 1/T$ 和角频率 $\omega = 2\pi/T$ 描述节奏。弹簧给出 $T = 2\pi\sqrt{m/k}$;单摆给出 $T = 2\pi\sqrt{L/g}$。周期与振幅无关,而每条公式都忽略一样东西——弹簧忽略引力,单摆忽略摆锤质量。