Conservation of Energy
Another way to derrive the energy equation is to use conservation
of energy. The diagram gets a little more complicated, but there
is no integration involved.
By conservation of energy we have
Notice that when the pendulum has swung into vertical position
and all the potential energy has been converted into kinetic energy
we have
We use this fact together with the observation that
and thus have
Now, we combine our equation for energy when the pendulum is vertical
(and thus achieves Vtmax) with the previous subsitution
for Δh and the substitution
to obtain
which of course is
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