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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 ...
A double pendulum consists of two pendulums attached end to end.. In physics and mathematics, in the area of dynamical systems, a double pendulum, also known as a chaotic pendulum, is a pendulum with another pendulum attached to its end, forming a simple physical system that exhibits rich dynamic behavior with a strong sensitivity to initial conditions. [1]
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 ...
Moreover, a double pendulum may exert motion without the restriction of only a two-dimensional (usually vertical) plane. In other words, the complex pendulum can move to anywhere within the sphere, which has the radius of the total length of the two pendulums. However, for a small angle, the double pendulum can act similarly to the simple ...
When the angle is small, the sine of is nearly equal to (see small-angle approximation), and so this expression simplifies to the equation for a simple harmonic oscillator with frequency = /. A harmonic oscillator can be damped, often by friction or viscous drag, in which case energy bleeds out of the oscillator and the amplitude of the ...
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.
Consider a planar pendulum with constant natural frequency ω in the small angle approximation. There are two forces acting on the pendulum bob: the restoring force provided by gravity and the wire, and the Coriolis force (the centrifugal force, opposed to the gravitational restoring force, can be neglected).
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.