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Hence, under the small-angle approximation, (or equivalently when ), = ¨ = where is the moment of inertia of the body about the pivot point . The expression for α {\displaystyle \alpha } is of the same form as the conventional simple pendulum and gives a period of [ 2 ] T = 2 π I O m g r ⊕ {\displaystyle T=2\pi {\sqrt {\frac {I_{O ...
In the case of a typical grandfather clock whose pendulum has a swing of 6° and thus an amplitude of 3° (0.05 radians), the difference between the true period and the small angle approximation (1) amounts to about 15 seconds per day.
In astronomy, the angular size or angle subtended by the image of a distant object is often only a few arcseconds (denoted by the symbol ″), so it is well suited to the small angle approximation. [6] The linear size (D) is related to the angular size (X) and the distance from the observer (d) by the simple formula:
The quantum pendulum is fundamental in understanding hindered internal rotations in chemistry, quantum features of scattering atoms, as well as numerous other quantum phenomena. Though a pendulum not subject to the small-angle approximation has an inherent nonlinearity, the Schrödinger equation for the quantized system can be solved relatively ...
Other phenomena can be modeled by simple harmonic motion, including the motion of a simple pendulum, although for it to be an accurate model, the net force on the object at the end of the pendulum must be proportional to the displacement (and even so, it is only a good approximation when the angle of the swing is small; see small-angle ...
The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that | | will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.
1718 Giulio Carlo Fagnano dei Toschi, studies the rectification of the lemniscate, addition results for elliptic integrals. [3]1736 Leonhard Euler writes on the pendulum equation without the small-angle approximation.
This is caused by the restoring force on the pendulum being circular not linear; thus, the period of the pendulum is only approximately linear in the regime of the small angle approximation. To be time-independent, the path must be cycloidal. To minimize the effect with amplitude, pendulum swings are kept as small as possible.