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Pendulum

A gravity pendulum is a weight on the end of a rigid or flexible line or rod, which, when given some initial lift from the vertical position, will swing back and forth under the influence of gravity over its central (lowest) point. A torsion pendulum consists of a body suspended by a fine wire or elastic fiber in such a way that it executes rotational oscillations as the suspending wire or fiber twists and untwists.

Gravity pendulums

For small displacements, the movement of an ideal pendulum can be described mathematically as simple harmonic motion, as the change in potential energy at the bottom of a circular arc is nearly proportional to the square of the displacement. Real pendulums do not have infinitesimal displacements, so their behavior is actually non-linear. Real pendulums will also lose energy as they swing, and so their motion will be damped, with the size of the oscillation decreasing approximately exponentially with time.

In the case of a pendulum with a point mass swinging on massless string or rod of length l, and an ambient gravity acceleration of g, the period of a complete oscillation is

This equation only applies when the amplitude of the swing is small. For a swing of , it is incorrect by only one part in 500. However, of course for a swing of it is maximally incorrect since this represents the bob balnced over its pivot point and so . A complete description of a pendulum's behavior is not mathematically simple.

Notice that the equation for T is independent of the bob's mass. This can be considered a consequence of the equivalence of gravitational mass and inertial mass: a heavier bob has a stronger restoring force, but on the other hand has more inertia and the two effects cancel.

The presence of g in the equation means that the pendulum frequency is different at different places on earth. So for example if you have an accurate pendulum clock in Glasgow (g=981.563 cm/s/s) and you take it to Cairo (g=979.317 cm/s/s), you must shorten the pendulum by 0.23%.

Two coupled pendulums form a double pendulum.

Torsion pendulums

If I is the moment of inertia of a body with respect to its axis of oscillation, and if K is the torsion coefficient of the fivre (torque required to twist it through an angle of one radian), then the period of oscillation of a torsion pendulum is given by

Both I and K may have to be determined by experiment. This can be done by measuring the period T and then adding to the suspended body another body of known moment of inertia I', giving a new period of oscillation T'

and then solving the two equations to get

The oscillating balance wheel of a watch is in effect a torsion pendulum, with the suspending fiber replaced by hairspring and pivots. The watch is regulated, first roughly by adjusting I (the purpose of the screws set radially into the rim of the wheel) and then more accurately by changing the free length of the hairspring and hence the torsion coefficient K.