Problems for Intermediate Methods in Theoretical Physics Edward F. Redish

Large Amplitude Pendulum with the Euler Methods

 Consider a mass, m , connected to a heavy rigid stand through a light (mass << m) but rigid rod as shown in the figure at the right. The motion of this mass is governed by the angular, motion form of Newton's second law, I dω/dt = τ, where I is the moment of inertia, τ is the torque, and ω is the angular velocity. This yields to the equations: (a) If the pendulum is started at t = 0 with θ(0) = θ0 and ω(0) = 0, use the Euler method to approximately find the angle of the bob at times tn = n Δt for n = 1, and 2, assuming Δt is small.

(b) In asking part (a), we said, "assuming Δt is small."   Compared to what?   Explain your choice.

(c) Commonly we make the approximation sin(θ) ≈ θ so that the equations become the same as that for the simple harmonic oscillator.   Use the Taylor series to estimate how big θ must be before this approximation is off by 10%.

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