Answer #338
The answer is (a): the period of the pendulum when swinging at a
large
angle is greater than when it is swung at a small angle, as seen
in an mpeg video by clicking your mouse on the photograph below.
By applying Newton's 2nd Law we have:
where the restoring force is the tangential component
of gravity. Notice how the restoring force is always towards the
center, and is thus negative.
The object's linear acceleration on
the other hand is obtained by implicit differentiation of the radian
measure of a circle twice. Take special note: this assumes the angular
displcement is measured in radians!
Or in other words, more succinctly:
which expresses the same as the above but in the generally accepted
form. Notice how the radius, r, refers
to the variable L in our situation.
Thus, exact (differential) equation for the pendulum is:
which is a slightly unwieldy equation that requires rather
sophisticated mathematics (see below). Since this is physics, the
above equation is instead usually simplified for the case of small
angles:
to become:
which is a rather charming homogeneous, ordinary differential equation of
second order.
By inspection, one can see that as the angle becomes larger, the
restoring force on the bob becomes relatively less: 
. Now, if the
restoring force is less, then the pendulum will not be drawn towards the
center as strongly, thus leading to a longer period for larger angles of
swing. This increase is about 6% for an angle of one radian.
For the brave of heart, the solution of the complete equation for the
pendulum, involving
large angles, calls for the use of elliptic integrals. For the
somewhate brave, the integral
however can be approximated (yet again) this time through use of infinite
series, of which the first three terms are:
where T is the exact period of the pendulum.
So in the interests of simplicity the
equation is simplified as described above, leading to the simple
pendulum.
Most intermediate mechanics textbooks discuss the complete solution of
the pendulum equation; a number of the references in our demonstration
G1-01: EXAMPLES OF SIMPLE HARMONIC MOTION also deal with the solution of
the more accurate equation.
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