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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:
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 the small-angle approximation, the motion of a simple pendulum is approximated by simple harmonic motion. The period of a mass attached to a pendulum of length l with gravitational acceleration g {\displaystyle g} is given by T = 2 π l g {\displaystyle T=2\pi {\sqrt {\frac {l}{g}}}}
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.
For small angles θ, cos(θ) ≈ 1; in which case so that for small angles the period t of a conical pendulum is equal to the period of an ordinary pendulum of the same length. Also, the period for small angles is approximately independent of changes in the angle θ. This means the period of rotation is approximately independent of the force ...
Geometrical optics is often simplified by making the paraxial approximation, or "small angle approximation". The mathematical behavior then becomes linear , allowing optical components and systems to be described by simple matrices.
Kater found that making one of the pivots adjustable caused inaccuracies, making it hard to keep the axis of both pivots precisely parallel. Instead he permanently attached the knife blades to the rod, and adjusted the periods of the pendulum by a small movable weight (b,c) on the pendulum shaft. Since gravity only varies by a maximum of 0.5% ...