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 |
| hertz/hɜːts/ | 赫兹 | hè zī |
| amplitude/ˈæmplɪtjuːd/ | 振幅 | zhèn fú |
A grandfather clock keeps time with a swing
- Inside a grandfather clock, a pendulum ticks back and forth at a perfectly steady rate.
- Each full swing takes the same time, whether it swings wide or narrow.
- That steady repeat time is the period 周期; how many per second is the frequency 频率.
- Getting them from the physics is the goal of this lesson.
落地摆钟用摆动来计时
- 在落地摆钟里,一个摆以完美稳定的速率来回滴答。
- 每一次完整摆动用时相同,无论摆得宽还是窄。
- 那个稳定的重复时间是周期;每秒有多少次是频率。
- 从物理中求出它们就是本节的目标。
Period and frequency
- The period $T$ is the time for one complete cycle (in seconds).
- The frequency $f$ is the number of cycles per second, in hertz 赫兹 ($\text{Hz}$).
- They are reciprocals: $f = \dfrac{1}{T}$.
- A fast oscillator has a short period and a high frequency.
周期与频率
- 周期 $T$ 是一个完整循环所用的时间(单位秒)。
- 频率 $f$ 是每秒的循环数,单位赫兹($\text{Hz}$)。
- 它们互为倒数:$f = \dfrac{1}{T}$。
- 快的振子周期短、频率高。
An oscillator has a period of $0.25\ \text{s}$. What is its frequency, in $\text{Hz}$? · 振荡器的周期为 $0.25\ \text{s}$。频率是多少，单位为 $\text{Hz}$？
$f = 1/T = 1/0.25 = 4\ \text{Hz}$.
Frequency is measured in . · 频率的单位是。
Frequency is in hertz ($\text{Hz}$) — cycles per second. · 频率的单位是赫兹（$\text{Hz}$）——即每秒振动的次数。
The two key formulas
- Mass on a spring: $T = 2\pi\sqrt{\dfrac{m}{k}}$ — heavier mass or softer spring means a slower swing.
- Simple pendulum: $T = 2\pi\sqrt{\dfrac{L}{g}}$ — a longer pendulum swings more slowly.
- Notice the pendulum's period does not depend on the mass of the bob.
- Both formulas come straight from the SHM equations.
两个关键公式
- 弹簧上的质量:$T = 2\pi\sqrt{\dfrac{m}{k}}$——更重的质量或更软的弹簧意味着更慢的摆动。
- 单摆:$T = 2\pi\sqrt{\dfrac{L}{g}}$——更长的摆摆得更慢。
- 注意摆的周期不取决于摆球的质量。
- 两个公式都直接来自 SHM 方程。
Length sets the period · 长度决定周期
Change the pendulum length (and gravity) and watch the period change, but not with amplitude. · 改变单摆长度（及重力）并观察周期的变化，但它不随振幅变化。
The period of a simple pendulum depends on: · 单摆的周期取决于：
$T = 2\pi\sqrt{L/g}$ — only length and gravity, not the bob's mass. · $T = 2\pi\sqrt{L/g}$ —— 仅长度和重力，而非摆锤质量。
You lengthen a pendulum. Its period: · 当增加单摆的长度时，其周期：
$T = 2\pi\sqrt{L/g}$: a larger $L$ gives a larger $T$ — a slower swing. · $T = 2\pi\sqrt{L/g}$：更大的$L$会导致更大的$T$——意味着摆动更慢。
Select all · 所有 quantities that change the period of a mass–spring oscillator. · 选择所有会改变弹簧振子周期的物理量。
$T = 2\pi\sqrt{m/k}$ depends on $m$ and $k$, not amplitude or colour. · $T = 2\pi\sqrt{m/k}$取决于$m$和$k$，与振幅或颜色无关。
Amplitude does not matter
- Remarkably, the period does not depend on the amplitude 振幅 (how far it swings).
- A wide swing travels farther but also moves faster, so the time is the same.
- This is why pendulum clocks keep good time even as the swing slowly shrinks.
- We call this property isochronous — "equal time".
振幅无关紧要
- 值得注意的是,周期不取决于振幅(摆得多远)。
- 大幅摆动走得更远,但也动得更快,所以时间相同。
- 这就是为什么摆钟即使摆幅慢慢变小也能保持准时。
- 我们把这个性质叫作等时性——"相等的时间"。
For SHM, swinging with a larger amplitude makes the period longer. · 对于SHM，用更大的振幅摆动会使周期变长。
The period of SHM is independent of amplitude — a wider swing is also faster, so the time is unchanged. · SHM的周期与振幅无关——更宽的摆动也更快，因此时间不变。
For SHM, the period is independent of amplitude — swinging wider does not take longer. And a pendulum's period does not depend on the mass of the bob, only on its length and $g$. These surprising facts trip up many students.
对 SHM,周期与振幅无关——摆得更宽不会用更长时间。而且摆的周期不取决于摆球的质量,只取决于它的长度和 $g$。这些出人意料的事实绊倒了许多学生。
A $0.5\ \text{kg}$ mass hangs on a spring with $k = 200\ \tfrac{\text{N}}{\text{m}}$. Find the period. (Use $\pi \approx 3.14$.)
- $T = 2\pi\sqrt{\dfrac{m}{k}} = 2\pi\sqrt{\dfrac{0.5}{200}} = 2\pi\sqrt{0.0025} = 2\pi(0.05) \approx 0.31\ \text{s}$.
Its frequency is $f = 1/T \approx 3.2\ \text{Hz}$.
一个 $0.5\ \text{kg}$ 的质量挂在 $k = 200\ \tfrac{\text{N}}{\text{m}}$ 的弹簧上。求周期。(取 $\pi \approx 3.14$。)
- $T = 2\pi\sqrt{\dfrac{m}{k}} = 2\pi\sqrt{\dfrac{0.5}{200}} = 2\pi\sqrt{0.0025} = 2\pi(0.05) \approx 0.31\ \text{s}$。
它的频率是 $f = 1/T \approx 3.2\ \text{Hz}$。
The period $T$ is the time for one cycle; the frequency $f = 1/T$ (in hertz) is cycles per second. A mass–spring has $T = 2\pi\sqrt{m/k}$; a pendulum has $T = 2\pi\sqrt{L/g}$. The period is independent of amplitude (isochronous).
周期 $T$ 是一个循环的时间;频率 $f = 1/T$(单位赫兹)是每秒的循环数。弹簧质量有 $T = 2\pi\sqrt{m/k}$;摆有 $T = 2\pi\sqrt{L/g}$。周期与振幅无关(等时性)。