Wednesday:
After reviewing one of the homework problems due, we started investigating pendulums.
We decided that the best variables/coordinate system to use was a radial-tangential system
in which the angle θ served as our displacement from equilibrium which was given by an
angle of zero measured from vertically down. There were two forces – a tension which was
directed radially inward and a weight which was directed vertically downward and could be
broken into tangential and radial components. The tangential component of F=ma was
m atan = m l
d2 θ / dt2 = - m g sin θ
Using the small angle approximation sin(θ) ≈ θ, gave us
l
d2 θ / dt2 = - g θ
which by analogy to our spring analysis has the solution
θ(t) = θmax cos(ω t + φ )
where
ω = (g/l)1/2
with l the length of the
pendulum.
We also derived results for quantities like the maximum velocity, and the energy
vmax = l
θmax ω
E = m g l
(1 - cos θmax )
We said little about the radial direction, but the tension (inward) minus the
radial component of the weight (outward) would give rise to the centripetal
acceleration which is given by
v2/l directed inward.
