Summary of Conditions

We may summarize the conditions under which we are justified in neglecting rotations or tilts of the earth and accelerations in directions other than that in the direction of freedom of the pendulum at rest.

Note that we have first restricted ourselves to hinged pendulums with motion constrained in a plane and have required the angular displacement to be small. Here,

  • T0

    is the free period of the pendulum.

  • L

    the distance from the axis of rotation to the center of oscillation (the equivalent simple pendulum length).

  • a

    that to the center of mass.

  • g

    is the acceleration of gravity.

  • T

    is the period of simple harmonic earth waves.

  • Λ

    is their wave length.

  • V

    is their apparent surface speed.

  • A

    is their amplitude.

 
gT022πL<<ΛA
(I)
 
T02L<<T2A
(II)
 
2πT0<<TΛA
(III)
 
g2π<<ΛT2=VT
(IVa)
 
g<<Λ2AT2=V2A
(IVb)
 
2πL<<Λ
(V)
 
4π2a<<Λ2A
(VI)

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