A simple pendulum. An idealized simple pendulum is given by a mass m hanging from a massless
string of length L. Its motion is described by the angle θ(t), where t is time. The distance travelled
by the pendulum is the arclength s(t) = Lθ(t). According to Newton’s 2nd law, the pendulum
mass×acceleration equals the restoring force Fnet acting on it.
mLθ00(t) = Fnet
where Fnet = mg sin θ (see picture). For small angles one often uses the approximation sin θ ≈ θ to
replace this differential equation by the simpler and easily solvable equation
mLθ00(t) = mgθ
Question: Consider the approximation sin θ ≈ θ. If the angle swings with −π/10 ≤ θ ≤ π/10, what
is an upper bound for the error made in this approximation?