Pendulum
This article is about gravity pendulums and torsion pendulums.
Pendulum has several other meanings:
- Pendulum is the title of several movies, including:
- Pendulum, a 1968 movie, directed by George Schaefer, starring George Peppard and Jean Seberg.
- Pendulum is the title of a 1970 album by Creedence Clearwater Revival.
- Pendulum is a trade name for a preemergent herbicide used for control of crabgrass in turf. Its active ingredient is pendimethalin.
A gravity pendulum is a weight on the end of a rigid 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.
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2 Torsion pendulums 3 See also |
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 behaviour is actually of a non-linear kind. 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 a massless rigid rod of length l, where is the angle between the rod and the vertical, the acceleration is given by and is equal to the angular acceleration multiplied by the length of the rod. So we have the following differential equation:
Gravity pendulums
When the amplitude of the swing is small, .
If the pendulum is initially still at an angle which is also the maximum angle, the function which solves the differential equation above is the following harmonic law:
The term is a pulsation, which is equal to ,
where is the period of a complete oscillation (outward and return).
Since ,
the period of a complete oscillation can be easily found, and it is given by Huygens's law:
For a swing of the bob is balanced over its pivot point and so . Keep in mind the pendulum is made of a rigid rod.
The following table shows the correction of the period T for wide amplitudes.
| 0 | 1.0000 |
| 15 | 1.0043 |
| 30 | 1.0174 |
| 45 | 1.0399 |
| 60 | 1.0732 |
| 75 | 1.1189 |
| 90 | 1.1803 |
| 105 | 1.2622 |
| 120 | 1.3729 |
| 135 | 1.5279 |
| 150 | 1.7622 |
| 165 | 2.1854 |
| 180 | |
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.
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
Torsion pendulums
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.