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The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to develop simplified equations like the following, where y is the distance of a fringe from the center of maximum light intensity, m is the order of the fringe, D is the distance between the slits and projection screen ...
Deviation of the "true" period of a pendulum from the small-angle approximation of the period. "True" value was obtained numerically evaluating the elliptic integral. Figure 4. Relative errors using the power series for the period. Figure 5. Potential energy and phase portrait of a simple pendulum.
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 half-angle formula for cosine can be obtained by replacing with / and taking the square-root of both sides: (/) = (+ ) /. Sine power-reduction formula: an illustrative diagram. The shaded blue and green triangles, and the red-outlined triangle E B D {\displaystyle EBD} are all right-angled and similar, and all contain the angle θ ...
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 ...
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 ...
The fundamental rectangle in the complex plane of . There are twelve Jacobi elliptic functions denoted by (,), where and are any of the letters , , , and . (Functions of the form (,) are trivially set to unity for notational completeness.) is the argument, and is the parameter, both of which may be complex.
In the small-angle approximation, the potential energy of a single pendulum system is , where g is the standard gravity, L is the length of the pendulum, m is the mass of the pendulum, and x is the horizontal displacement of the pendulum.