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

The angle is 4.44º

Step-by-step explanation:

If:

`(sin\theta -\theta)/(\theta ) \leq 0.1o/o\\(sin\theta -\theta)/(\theta )\leq 10^(-3)`

According the Taylor`s series:

`sin\theta =\theta -(\theta ^(3) )/(3!) ....`

`(\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`