Simple Pendulum

A simple pendulum is one which can be considered to be a point mass suspended from a string or rod of negligible mass. It is a resonant system with a single resonant frequency. For small amplitudes, the period of such a pendulum can be approximated by:

Show 显示 For pendulum length L = cm = m 对于单摆长度 L = cm = m 中文译文中的待填/计算数值依次对应：1：L 2：Lm。实际数值以上方原输入框为准。 and acceleration of gravity g = m/s2 加速度为重力加速度 g = m/s² 中文译文中的待填/计算数值依次对应：1：g。实际数值以上方原输入框为准。 the pendulum period is T = s 单摆的周期是 T = s 中文译文中的待填/计算数值依次对应：1：t。实际数值以上方原输入框为准。 (Enter data for two of the variables and then click on the active text for the third variable to calculate it.) Note that the angular amplitude does not appear in the expression for the period. This expression for period is reasonably accurate for angles of a few degrees, but the treatment of the large amplitude pendulum is much more complex. 需要注意的是，角振幅并不出现在周期的表达式中。这个周期表达式对于角度在几度范围内的情况是合理准确的，但对于大振幅的单摆处理则要复杂得多。 输入两个变量的数据，然后点击第三个变量的活性文本以计算它。

If the rod is not of negligible mass, then it must be treated as a physical pendulum. 如果杆的质量不可忽略，就必须将其视为物理摆。

Sean Carroll relates the story of Galileo's discovery of the fact that for small amplitudes, the period and frequency are unaffected by the amplitude. "In 1581, a young Galileo Galilei reportedly made a breakthrough discovery while he sat bored during a church service in Pisa. The chandelier overhead would swing gently back and forth, but it seemed to move more quickly when it was swinging widely (after a gust of wind, for example) and more slowly when it wasn't moving as far. Intrigued, Galileo decided to measure how much time it took for each swing, using the only approximately periodic event to which he had ready access: the beating of his own pulse. He found something interesting: The number of heartbeats between swings of the chandelier was roughly the same, regardless of whether the swings were wide or narrow. The size of the oscillations - how far the pendulum swung back and forth - didn't affect the frequency of those oscillations."

Pendulum Motion

The motion of a simple pendulum is like simple harmonic motion in that the equation for the angular displacement is

Show 显示 which is the same form as the motion of a mass on a spring: 这与弹簧上质量的运动形式相同： The anglular frequency of the motion is then given by 该运动的角频率则由 compared to 与...相比 for a mass on a spring. 对于弹簧上的质量。

The frequency of the pendulum in Hz is given by 摆的频率（单位：Hz）由

and the period of motion is then 则运动周期为

Period of Simple Pendulum

A point mass hanging on a massless string is an idealized example of a simple pendulum. When displaced from its equilibrium point, the restoring force which brings it back to the center is given by:

Show 显示 For small angles θ, we can use the approximation 对于小角度θ，我们可以使用近似 Show 显示 in which case Newton's 2nd law takes the form 在这种情况下，牛顿第二定律的形式为 Even in this approximate case, the solution of the equation uses calculus and differential equations. The differential equation is 即使在这种情况的近似情况下，方程的解仍需要用到微积分和微分方程。微分方程是 and for small angles θ the solution is: 并且对于小角度θ，解为:

Pendulum Geometry

The period of a simple pendulum for small amplitudes θ is dependent only on the pendulum length and gravity. For the physical pendulum with distributed mass, the distance from the point of support to the center of mass is the determining "length" and the period is affected by the distribution of mass as expressed in the moment of inertia I.

Pendulum Equation

The equation of motion for the simple pendulum for sufficiently small amplitude has the form

which when put in angular form becomes 当将其转换为角形式时变为 This differential equation is like that for the simple harmonic oscillator and has the solution: 这个微分方程与简谐振子的方程类似，其解为：