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# 11.3: Torsion Pendulum

[ "article:topic", "authorname:tatumj", "showtoc:no" ]

A torsion pendulum consists of a mass of rotational inertia $$I$$ hanging by a thin wire from a fixed point. If we assume that the torque required to twist the wire through an angle $$\theta$$ is proportional to $$\theta$$ and to no higher powers, then the ratio of the torque to the angle is called the torsion constant $$c$$. It depends on the shear modulus of the material of which the wire is made, is inversely proportional to its length, and, for a wire of circular cross-section, is proportional to the fourth power of its diameter. A thick wire is much harder to twist than a thin wire. Ribbonlike wires have comparatively small torsion constants. The work required to twist a wire through an angle $$\theta$$ is $$\frac{1}{2}c\theta^{2}$$.

When a torsion pendulum is oscillating, its Equation of motion is

$I\ddot{\theta}=-c\theta. \label{11.3.1}$

This is an Equation of the form 11.1.5 and is therefore simple harmonic motion in which $$\omega=\sqrt{\frac{c}{I}}$$. This example, incidentally, shows that our second definition of simple harmonic motion (i.e. motion that obeys a differential Equation of the form of Equation 11.1.5) is a more general definition than our introductory description as the projection upon a diameter of uniform motion in a circle. In particular, do not imagine that $$\omega$$ here is the same thing as $$\dot{\theta}$$!

Exercise $$\PageIndex{1}$$

Write down the torsional analogues of all the Equations given for linear motion in Sections 11.1 and 11.2.