Chapter 16, Problem 28PQ

We do not need the analogy in Equation 16.30 to write expressions for the translational displacement of a pendulum bob along the circular arc s(t), translational speed v(t), and translational acceleration a(t). Show that they are given by

s(t) = s_max cos (ω_smpt + φ)

v(t) = −v_max sin (ω_smpt + φ)

a(t) = −a_max cos(ω_smpt + φ)

respectively, where s_max = ℓθ_max with ℓ being the length of the pendulum, v_max = s_max ω_smp, and a_max = s_max ${\omega }_{\text{smp}}^2$.