Simple Pendula

(48) |

where is the moment of inertia of the mass, and the torque acting about the suspension point. For the case in hand, given that the mass is essentially a point particle, and is situated a distance from the axis of rotation (i.e., from the suspension point), it follows that (ibid.).

The two forces acting on the mass are the downward gravitational force,
, where
is the acceleration due to gravity,
and the tension,
, in the string.
However, the tension makes no contribution to the torque,
because its line of action passes
through the suspension point. From elementary trigonometry,
the line of action of the gravitational force passes a perpendicular distance
from the
suspension point. Hence, the magnitude of the gravitational torque is
.
Moreover, the gravitational torque is a *restoring torque*: that is, if
the mass is
displaced slightly from its equilibrium position (i.e.,
) then the
gravitational torque acts
to push the mass back towards that position. Thus, we can write

(49) |

Combining the previous two equations, we obtain the following angular equation of motion of the pendulum,

Unlike all of the other time evolution equations that we have examined, so far, in this chapter, the preceding equation is nonlinear [because ], which means that it is generally very difficult to solve.

Suppose, however, that the system does not stray very far from its equilibrium position ( ). If this is the case then we can expand in a Taylor series about . (See Appendix B.) We obtain

(51) |

If is sufficiently small then the series is dominated by its first term, and we can write . This is known as the

where

(53) |

Equation (52) is the simple harmonic oscillator equation. Hence, we can immediately write its solution in the form

where and are constants. We conclude that the pendulum swings back and forth at a fixed angular frequency, , that depends on and , but is independent of the amplitude, , of the motion. This result only holds as long as the small angle approximation remains valid. It turns out that is a good approximation provided . Hence, the period of a simple pendulum is only amplitude independent when the (angular) amplitude of its motion is less than about .