(Tilted pendulum) Consider a rod of length L with a point mass m at its end, where the mas...

(Tilted pendulum) Consider a rod of length L with a point mass m at its end, where the mass of the rod is negligible com­pared to m. The rod is welded at a right angle to another, which rotates without friction about an axis that is tilted by an angle of α with respect to the vertical (see figure). Let θ denote

the angle of rotation of the pendulum, with respect to its equilibrium position (where m is at its lowest possible point, namely, in the plane of the paper).

Derive the governing equation of motion

As a partial check of this result, observe that for α = Π/2 (14.1) does reduce to the equation of motion of the ordinary pendulum (see Exercise). HINT: Write down an equation of conservation of energy (kinetic plus potential energy equal a constant), and differentiate it with respect to the time t.

Exercise

(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