I was conducting an experiment involving the effect of large angular displacements (10°, 20°, ... 90°) on the amplitude of a simple pendulum. So far, I have found that the period of the pendulum varies according this equation:  \begin{align}T&amp;=  2π\sqrt{\frac {l}{g}} \left(1 + \frac{1}{16} θ^2 + \frac{11}{3072} θ^4  + \frac{173}{737280} θ^6  + \frac{22931}{1321205760} θ^8\right. \\ &amp;\left.\qquad\qquad+   \frac{1319183}{951268147200} θ^{10} + \frac{233526463}{2009078326886400} θ^{12} + …\right) \end{align}  Essentially, period increases with increasing angular displacement. However, I don't really understand why the period varies in this way. Could someone explain this to me without the use of math (no small angle approximations/Taylor series)?   For example, in this way: Although the pendulum accelerates for a longer period of time and has a greater maximum velocity, the additional distance it has to travel is greater than what its larger acceleration and maximum speed can make up for (this is just my guess).
Suppose you have a physical . It is true that as amplitude increases, the period increases. Can we demonstrate this fact without explicitly finding the period (which is pretty involved and pretty messy) in:    an intuitive fashion,  rigorously?
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Back in the day of TSR (Terminate and Stay Resident) MS-DOS programs this was possible. Certainly it must be in Ubuntu too?  Do I have to write a daemon to accomplish this?    I found other questions with no answers: ,   I found a similar question with an answer: . However that question stipulated the answer could not be AutoKey. What is the difference between "INNER JOIN" and "OUTER JOIN"? Is circular reasoning always a fallacy?