Question

The restoring force on a pendulum bob is proportional to sin(θ), where θ is the angle which the pendulum makes with the vertical. The pendulum performs simple harmonic motion if the approximation sin(θ) ≈ θ is satisfied. Which angle below is the largest θ can be while the approximation holds to within 0.1%?

Answer 1

Answer:

The angle is 4.44º

Explanation:

If:

$\frac{sin\theta -\theta}{\theta } \leq 0.1o/o\\\frac{sin\theta -\theta}{\theta }\leq 10^{-3}$

According the Taylor`s series:

$sin\theta =\theta -\frac{\theta ^{3} }{3!} ....$

$\frac{\theta ^{2} }{6} \leq 10^{-3} \\\theta ^{2} \leq 6x10^{-3} \\\theta \leq \sqrt{6x10^{-3} } \\\theta \leq 0.0775\\\theta = 0.0775rad=4.44$