Simple and Physical Pendulums · 简单单摆与物理单摆
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| simple pendulum/ˈsɪmpl ˈpendjʊləm/ | 单摆 | dān bǎi |
| physical pendulum/ˈfɪzɪkl ˈpendjʊləm/ | 物理摆 | wù lǐ bǎi |
Every swing keeps time
- A grandfather clock's rod ticks out steady seconds.
- A child on a swing takes the same time whether pushed high or low.
- A swinging body -- thin or bulky -- has its own fixed beat.
- Two formulas time any small swing about a pivot.
每一次摆动都在计时
- 落地大钟的摆杆稳稳地敲出秒。
- 秋千上的孩子无论被推得高还是低,花的时间都一样。
- 摆动的物体——无论细还是粗——都有自己固定的节拍。
- 两条公式为绕枢轴的任何小摆动计时。
The simple pendulum
- A simple pendulum 单摆 is a point mass on a light string of length $L$:
- Gravity provides a restoring torque that grows with the angle.
- For small angles the swing is SHM.
单摆
- 单摆是挂在长度为 $L$ 的轻绳上的质点:
- 重力提供随角度增大的回复力矩。
- 在小角度下,摆动是简谐运动。
A simple pendulum has $L = 0.25\ \text{m}$ and $g = 9.8\ \text{m/s}^2$. Its period (in s, to 1 decimal)? · 单摆具有$L = 0.25\ \text{m}$和$g = 9.8\ \text{m/s}^2$。其周期（单位：s，保留1位小数）是多少？
$T = 2\pi\sqrt{L/g} = 2\pi\sqrt{0.25/9.8} \approx 1.0\ \text{s}$.
Doubling the bob mass of a simple pendulum changes its period. · 增加简单单摆摆锤的质量会改变其周期。
$T = 2\pi\sqrt{L/g}$ has no mass -- the bob's mass cancels. · $T = 2\pi\sqrt{L/g}$中没有质量——摆锤的质量被抵消。
The physical pendulum
- A physical pendulum 物理摆 is any rigid body swinging about a pivot:
- $I$ is the rotational inertia about the pivot, $d$ the pivot-to-centre distance.
- A ruler, a bat, a leg -- each swings with this period.
物理摆
- 物理摆是绕枢轴摆动的任意刚体:
- $I$ 是绕枢轴的转动惯量,$d$ 是枢轴到质心的距离。
- 一把尺、一根球棒、一条腿——都以这个周期摆动。
In the physical pendulum period $T = 2\pi\sqrt{I/(mgd)}$, what is $d$? · 在物理单摆周期公式$T = 2\pi\sqrt{I/(mgd)}$中，$d$是什么？
$d$ is the pivot-to-centre-of-mass distance -- the lever arm of gravity. · $d$是支点-质心距离——重力的力臂。
One is a special case of the other
- For a point mass at distance $L$: $I = mL^2$ and $d = L$.
- Plug in: $T = 2\pi\sqrt{mL^2/(mgL)} = 2\pi\sqrt{L/g}$.
- So the simple pendulum is just the physical pendulum's simplest case.
一个是另一个的特例
- 对于距离 $L$ 处的质点:$I = mL^2$,$d = L$。
- 代入:$T = 2\pi\sqrt{mL^2/(mgL)} = 2\pi\sqrt{L/g}$。
- 所以单摆只是物理摆最简单的情形。
Simple and physical pendulums · 简单单摆与物理单摆
Change the length and gravity and see how the pendulum's period responds - amplitude barely matters. · 改变长度和重力，观察单摆周期的响应——振幅几乎无关紧要。
The simple-pendulum formula is a special case of the physical-pendulum formula. · 简单单摆公式是物理单摆公式的特例。
Put $I = mL^2$ and $d = L$ into · 生成 $T = 2\pi\sqrt{I/(mgd)}$ and you recover $2\pi\sqrt{L/g}$. · 将$I = mL^2$和$d = L$代入$T = 2\pi\sqrt{I/(mgd)}$即可恢复$2\pi\sqrt{L/g}$。
A uniform rod of length $L$ pivoted at one end has $I = \tfrac{1}{3}mL^2$ and $d = L/2$.
- $T = 2\pi\sqrt{\dfrac{I}{mgd}} = 2\pi\sqrt{\dfrac{\tfrac{1}{3}mL^2}{mg\cdot L/2}} = 2\pi\sqrt{\dfrac{2L}{3g}}$.
- It swings a little faster than a simple pendulum of the same length.
一根长度为 $L$、在一端为枢轴的均匀杆,有 $I = \tfrac{1}{3}mL^2$,$d = L/2$。
- $T = 2\pi\sqrt{\dfrac{I}{mgd}} = 2\pi\sqrt{\dfrac{\tfrac{1}{3}mL^2}{mg\cdot L/2}} = 2\pi\sqrt{\dfrac{2L}{3g}}$。
- 它比同样长度的单摆摆得稍快一些。
For a physical pendulum, the rotational inertia $I$ must be measured about the... · 对于物理单摆，转动惯量$I$必须关于...测量
The body rotates about the pivot, so $I$ is taken about that axis. · 物体绕支点旋转，因此$I$取关于该轴的值。
Both pendulum period formulas are valid only when the swing angle is... · 两种单摆周期公式仅在摆角为...时有效
Only small angles make the restoring torque proportional to displacement (SHM). · 只有小角度才能使回复力矩与位移成正比（简谐运动）。
Both formulas assume small angles -- beyond about $15^\circ$ the motion is no longer true SHM. For a physical pendulum, take $I$ about the pivot (use the parallel-axis theorem if you know $I$ about the centre), and let $d$ be the pivot-to-centre-of-mass distance, not the full length. The bob's mass cancels for a simple pendulum but need not for a physical one.
两条公式都假设小角度——超过约 $15^\circ$,运动就不再是真正的简谐运动。对物理摆,$I$ 要绕枢轴来取(若已知绕质心的 $I$,用平行轴定理),而 $d$ 是枢轴到质心的距离,不是全长。单摆的摆锤质量会约掉,物理摆则未必。
A simple pendulum (point mass on a string) swings with $T = 2\pi\sqrt{L/g}$; a physical pendulum (any rigid body on a pivot) swings with $T = 2\pi\sqrt{I/(mgd)}$, where $I$ is taken about the pivot and $d$ reaches the centre of mass. The simple case falls out of the physical one, and both are SHM only for small angles.
单摆(绳上的质点)以 $T = 2\pi\sqrt{L/g}$ 摆动;物理摆(枢轴上的任意刚体)以 $T = 2\pi\sqrt{I/(mgd)}$ 摆动,其中 $I$ 绕枢轴取,$d$ 到达质心。简单情形从物理摆中导出,而两者都只在小角度时是简谐运动。