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51. If the torque exerted by the spring on the rod is proportional to the angle of rotation of
the rod and if the torque tends to pull the rod toward its equilibrium orientation, then the
rod will oscillate in simple harmonic motion. If
τ
= –
C
θ
, where
is the torque,
is the
angle of rotation, and
C
is a constant of proportionality, then the angular frequency of
oscillation is
ω
=
CI
/
and the period is
TI
C
==
22
ππ
//
,
where
I
is the rotational inertia of the rod. The plan is to find the torque as a function of
and identify the constant
C
in terms of given quantities. This immediately gives the
period in terms of given quantities. Let
A
0
be the distance from the pivot point to the wall.
This is also the equilibrium length of the spring. Suppose the rod turns through the angle
, with the left end moving away from the wall. This end is now (
L
/2) sin
further from
the wall and has moved a distance (
L
/2)(1 – cos
) to the right. The length of the spring is
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This note was uploaded on 06/03/2011 for the course PHY 2049 taught by Professor Any during the Spring '08 term at University of Florida.
 Spring '08
 Any
 Physics, Simple Harmonic Motion

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