11 OSCILLATORY MOTION 11.3 The torsion pendulum
±
torsion wirediskfixed supportFigure 96: A torsion pendulum.since m ω^2 = k and sin^2 θ + cos^2 θ = 1. Note that the total energy is a constant of
the motion, as expected for an isolated system. Moreover, the energy is propor-
tional to the amplitude squared of the motion. It is clear, from the above expres-
sions, that simple harmonic motion is characterized by a constant backward and
forward flow of energy between kinetic and potential components. The kinetic
energy attains its maximum value, and the potential energy attains it minimum
value, when the displacement is zero (i.e., when x = 0 ). Likewise, the potential
energy attains its maximum value, and the kinetic energy attains its minimum
value, when the displacement is maximal (i.e., when x = a). Note that the
minimum value of K is zero, since the system is instantaneously at rest when the
displacement is maximal.
11.3 The torsion pendulum
Consider a disk suspended from a torsion wire attached to its centre. See Fig. 96.
This setup is known as a torsion pendulum. A torsion wire is essentially inexten-
sible, but is free to twist about its axis. Of course, as the wire twists it also causes
the disk attached to it to rotate in the horizontal plane. Let θ be the angle of
rotation of the disk, and let θ = 0 correspond to the case in which the wire is
untwisted.
Any twisting of the wire is inevitably associated with mechanical deformation.The wire resists such deformation by developing a restoring torque, τ, which acts