Most seismometers are of the pendulum type, i.e. they let the mass rotate around an axis rather than move along a straight line (Figs. 7 to 10; the point bearings in our figures are for illustration only, most seismometers have crossed flexural hinges). Pendulums are not only sensitive to translational but also to angular acceleration. Since the rotational component in seismic waves is normally small, there is not much practical difference between linear-motion and pendulum-type seismometers. However, they may behave differently in technical applications or on a shake table where it is not uncommon to have noticeable rotations.
For small translational ground motions, the equation of motion of a pendulum is formally identical to Eq. (25) but z must then be interpreted as the angle of rotation. Since the rotational counterparts of the constants M, R, and S in Eq. (25) are of little interest in modern electronic seismometers, we will not discuss them further and refer the reader instead to the older literature, such as [Berlage Jr. 1932] or [Willmore, P. L. (ed.) 1979].
The simplest example of a pendulum is a mass suspended with a string or wire (like Foucault's pendulum). When the mass has a small size compared to the length of the string so that it can be idealized as a point mass, then this is called a mathematical pendulum. Its period of oscillation is where g is the gravitational acceleration. A mathematical pendulum of 1 m length has a period of nearly 2 seconds; for a period of 20 seconds the length has to be 100 m. Clearly, this is not a suitable design for a long-period seismometer.