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Calculus/Pendulum Wall Clock.


Dear Prof Mervyn

Is it possible to design and Construct a Pendulum Wall Clock where the pendulum can rotate 180 degree?.


This is the real mathematics of pendulum motion based on elliptic functions.
  Denote by W the weight in lb. of the pendulum and let OG = h(feet), where G is the centre of gravity ; let Wk^2 denote the moment of inertia of the pendulum about the horizontal axis through O

 Then if OG makes an angle x with the vertical, then from elementary mechanics

   W(h^2 + k^2)d^2x/dt^2 = -Wghsinx
Finally from elementary calculus and mechanics we get .5l( dx/dt)^2 = C- gversx

On using Jacob's theory of elliptic functions , some elementary mathematical manipulations, the Binomial Theorem and Wallis's Theorem we period of a pendulum length l, oscillating through an angle 2a, is

T= 4K/n = 2pi[(l/g{1 + (1/2)^2(sin(a/2)^2 + (1/2*3/4)^2(sin1/2a)^4 + 1/2*3/4*5/6)^2(sin1/2a)^6 + ...]^.5

Let 2a = pi radians then sin pi/2 = 1 and T becomes infinite.  


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I can solve problems in calculus, advanced calculus, differential and partial differential equations, complex variables, Fourier series and integrals, analysis,Gram-Schmidt orthogonalization, and Legendre polynomials. Soon I will be able to answer any question on advanced engineering control.


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