A trigonometric approximation for the tension in the string of a simple pendulum accurate for all amplitudes

In a previous work, O'Connell ( Phys. Teach. 40 , 24 (2002)) investigated the time dependence of the tension in the string of a simple pendulum oscillating within the small-angle regime. In spite of the approximation sin θ ≈ θ being accurate only for amplitudes below 7°, his experimental results are for a pendulum oscillating with an amplitude of about 18°, therefore beyond the small-angle regime. This lapse may also be found in some textbooks, laboratory manuals and internet. By noting that the exact analytical solution for this problem involves the so-called Jacobi elliptic functions , which are unknown to most students (even instructors), I take into account a sinusoidal approximate solution for the pendulum equation I introduced in a recent work ( Eur. J. Phys. 29 1091 (2008) ) for deriving a simple trigonometric approximation for the tension valid for all possible amplitudes. This approximation is compared to both the O'Connell and the exact results, revealing that it is accurate enough for analysing large-angle pendulum experiments.