The Simple Pendulum

A SIMPLE PENDULUM consists of a small, heavy bob of mass $m$ -- idealised as a single point mass, with all its mass concentrated at one point -- suspended by a light string (treated as perfectly massless) of length $L$ from a fixed support, free to swing back and forth in a single vertical plane.\n\nWhen the bob is pulled aside through a small angle $\theta$ from the vertical and then released, gravity supplies a RESTORING TORQUE about the point of suspension, tending to swing the bob back towards the vertical. Resolving the bob's weight $mg$ into components along the string and perpendicular to it, the perpendicular component -- the one actually responsible for the restoring effect -- is $mg\sin\theta$, so the restoring torque about the pivot has magnitude\n\n$\tau = -mgL\sin\theta$\n\n(the negative sign showing that the torque always acts to DECREASE $\theta$, i.e. to oppose whatever angular displacement currently exists).\n\nFor SMALL angular displacements -- conventionally, $\theta$ less than about $15^\circ$, with $\theta$ expressed in radians -- the small-angle approximation $\sin\theta \approx \theta$ may be used, so that the restoring torque simplifies to\n\n$\tau \approx -mgL\theta$\n\nUsing the rotational form of Newton's second law, $\tau = I\alpha = I\,d^2\theta/dt^2$, and noting that the moment of inertia of a single point mass $m$ located a distance $L$ from the pivot is simply $I = mL^2$, this becomes\n\n$mL^2\,\frac{d^2\theta}{dt^2} = -mgL\theta \quad\Longrightarrow\quad \frac{d^2\theta}{dt^2} = -\frac{g}{L}\,\theta$\n\nComparing this equation directly with the standard form of the S.H.M. equation, $d^2\theta/dt^2 = -\omega^2\theta$, immediately identifies\n\n$\omega = \sqrt{\frac{g}{L}} \qquad\Longrightarrow\qquad T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{L}{g}}$\n\nwhich is the familiar time-period formula for a simple pendulum. Notice, in this final result, that both the mass $m$ of the bob and the angular amplitude $\theta_0$ of the swing have completely cancelled out of the answer: the period depends ONLY on the pendulum's length $L$ and on the local value of $g$, and is the SAME whether the bob is light or heavy, and whether the swing is small or only slightly larger -- as long as the small-angle approximation still holds reasonably well.\n\nThis derivation rests on several assumptions that must all be kept firmly in mind, since the formula fails once any of them is seriously violated: the string is treated as perfectly massless and completely inextensible; the bob is treated as an ideal point mass, with all of its mass concentrated at a single point (a real, physically extended bob would instead need the more general treatment of a "physical pend …