(Grandfather clock) Consider a pendulum governed by the equation of motion mL6" + mg s...

(Grandfather clock) Consider a pendulum governed by the equation of motion mL6" + mg sin 0 = 0, or

where g is the acceleration of gravity. (See the figure.) If

| θ| ≪ 1 (where ≪ means much smaller than), then sin θ ≈ θ, and the nonlinear equation of motion (8.1) can be simplified to the linear equation

or, if we allow for some inevitable amount of damping due to friction and air resistance,

where 0