Simple Pendulums

Abstract

Ideal pendulums are idealisations of real pendulums that are often used in order to make mathematical analysis more tractable. Ideal pendulums are not physically realistic. In scientific terminology, an ideal pendulum is a model of a real pendulum. Results of analyses of ideal pendulums often provide useful approximations for real pendulums, such as clock pendulums. A simple pendulum is a particular type of ideal pendulum, and the term is often used without explanation. In a simple pendulum, the bob of a real clock pendulum is replaced by a particle whose dimensions are small enough for it to be adequately represented by a mathematical point. When a specified mass, m, is ascribed to a particle it is called a point mass. The only force on a simple pendulum is that due to gravity on the point mass. There are actually two types of simple pendulum. In a simple rod pendulum the rod of a real clock pendulum is replaced by a rigid, massless rod, length l, with a point mass m, replacing the bob, attached to the lower end. Rigid means that the rod does not deform under an applied load, in particular the length always stays the same. In other words the rod is inextensible. The suspension of a real pendulum, is replaced by a horizontal frictionless pivot, where frictionless means that the pivot does not exert any force as the rod rotates about the pivot, and the pendulum swings in a vertical plane. The frictionless pivot is the suspension point. This is fixed in space so the path of the point mass is a circular arc, radius l. It is usually sufficient to regard a point on the Earth’s surface as being fixed in space. A simple rod pendulum is not a chaotic system, so it does not display chaotic behaviour. In a simple string pendulum the string of a real pendulum, such as a pendulum for occult uses, is replaced by a massless string, length l, with a point mass m, replacing the bob, attached to the lower end. The string is inextensible in the sense that, when pulled taut, it is always the same length. The string has no resistance to bending and the suspension point is a clamp at the upper end. The clamp is fixed in space so, provided that the string remains taut, the path of the point mass is on a sphere, radius l. The path taken by the point mass depends on how the pendulum is launched. A simple string pendulum is a chaotic system that can display chaotic behaviour. Simple pendulums are ideal pendulums and, once launched, they will continue to swing indefinitely with the same amplitude. Simple pendulum are approximately isochronous for small amplitudes, and the motion of the point mass approximates to simple harmonic motion. Both types of simple pendulum are in stable equilibrium when in the rest position, which is vertically downwards. Stable equilibrium means that, following a small displacement, a simple pendulum tends to return to the rest position. If the simple rod pendulum is vertically upwards then it is in unstable equilibrium and, following a small displacement, the point mass does not return to its initial position. A simple string pendulum does not have an unstable equilibrium position. For some purposes it is not necessary to distinguish between the two types of simple pendulum, and authors often refer to ‘the simple pendulum’ without explanation.