For the small oscillations of a pendulum whose length \(l\) is varying very slowly show that the maximum angular displacement \(\theta_{\max } \propto l^{-3 / 4}\). Hence show that the maximum sideways displacement from the vertical is \(\propto l^{1 / 4}\) and that the maximum acceleration is \(\propto l^{-3 / 4}\). (Note that this last result implies, as \(l\) decreases slowly, an increasing risk for spaghetti eaters from sauce detachment!)

In the realm of pendulum motions, understanding angular displacement is crucial. Angular displacement, denoted by the Greek letter theta \(\theta\), refers to the angle through which the pendulum pivots from its vertical resting position during oscillation. Imagine the pendulum as a clock's hand at noon. As it swings, it moves like the minute hand, marking an angle from the top that increases until reaching its farthest extent, the maximum angular displacement, and then decreases back to zero as it swings to the other side.

For small oscillations, this displacement is intricately connected to the time of oscillation and the length of the pendulum, represented by \(l\). Precisely, \(\theta(t) = \theta_{\max}\cos(\sqrt{\frac{g}{l}} t)\), where \(\theta_{\max}\) represents the peak angular displacement at any given moment. Recognizing that angular displacement scales with the pendulum length as \(\theta_{\max} \propto l^{-\frac{3}{4}}\) unveils a fascinating behavior: the shorter the pendulum, the smaller the angle of sweep for the same energy input. This characteristic has far-reaching implications, from the tuning of delicate horological mechanisms to enhancing our understanding of dynamic systems.

Visualize the pendulum's swing, not just as an arc, but how far sideways it travels at its maximum reach. This is characterized as the maximum sideways displacement from the vertical, symbolized as \(x_{\max}\). Using the angular displacement \(\theta_{\max}\), we derive \(x_{\max}\) as \(x_{\max} = l\sin(\theta_{\max}))\).

Due to the small angle approximation for pendulum swings, \(\sin(\theta_{\max})\) can be substituted with \(\theta_{\max}\), which simplifies the equation to \(x_{\max} \approx l\theta_{\max}\). This simplification leads us to understand that \(x_{\max} \propto l^{\frac{1}{4}}\), suggesting that as the length of the pendulum increases, there is a gentle rise in the maximum sideways reach. This relationship is vital for designing pendulum-dependent systems such as seismometers or even children's swing sets, ensuring we appreciate the spatial requirements as lengths change.

Acceleration is a telling measure of a pendulum's dynamics. It is the rate at which the pendulum's speed increases as it hurtles toward its lowest point. Maximum acceleration is particularly noteworthy. It tells us how quickly the pendulum gains speed from rest at the extremities of its swing.

Through some calculus, involving the double derivation of the angular displacement formula with respect to time, we land on \(a_{\max} = -l\omega^2\theta_{\max}\) where \(\omega\) is the angular frequency. Hence, like a tightly-wound spring, as the pendulum becomes shorter (\(l\) decreases), \(a_{\max} \propto l^{-\frac{3}{4}}\). An intriguing, if comical, application of this is being careful while slurping spaghetti. As your fork twirls, a small pendulum forms. Too much vigor and the sauce may detach, as the maximum acceleration increases when \(l\) decreases. A light-hearted reminder that the physics of pendulums can affect even the most everyday activities!