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7.2.1 Derivation of the expression for the period of a Simple Pendulum
The particle in a simple pendulum moves in an arc of a circle of radius l. The forces acting on the particle are its weight mg and the tension T along the string. The tangential component of the resultant force is, from the figure,
FT = – mg sin q
where the minus sign appears because it is opposed to the displacement s = CA. The equation for the tangential motion is
FT = maT
Since the particle moves along a circle of radius l, we may use Equations, aT = R(d2q/dt2), with R replaced by l, to express the tangential acceleration. That is,
The equation for the tangential motion is thus
This equation is not of the same type as Equation because of the presence of sin q instead of just q. However, if the angle q is small, which is true if the amplitude of the oscillations is very small, we may use the series expansion of sin q and keep only the first term. So we write in equation for the motion of the pendulum, which becomes
This is an equation identical to Equation, with x replaced by q, this time referring to angular rather than linear motion. Thus we may conclude that, within our approximation, the angular motion of the pendulum is simple harmonic, with
The angle q can thus be expressed in the form
Using Equation, P = 2p/w, we can express the period of oscillation of the pendulum as
which is equation . For large amplitudes the approximation is not valid and the calculation of the period is more complex.
It is relatively easy and instructive to obtain a better approximation for P. Again using the series expansion of sin q (Appendix, Equation), keeping only the first two terms, we may write
which when inserted in Equation gives a second-order approximation to the equation of motion,
We may make another approximation by replacing q2 in the parentheses by . Then
which again is an equation of type. Therefore, up to this order of approximation
A more precise approximation gives
where the amplitude q0 must be expressed in radians. This is a sufficient approximation for most practical situations. In fact, the corrective term amounts to less than 1% for amplitudes less than 23° or 0.40 radians.
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