Rotating Pendulum

The bearing of a rigid pendulum of mass m is forced to rotate uniformly with angular velocity ω (see Figure 1.1). The angle between the rotation

Figure 1.1

 axis and the pendulum is called θ. Neglect the inertia of the bearing and of the rod connecting it to the mass. Neglect friction. Include the effects of the uniform force of gravity.

a) Find the differential equation for θ.

b) At what rotation rate ω_c does the stationary point at θ = 0 become unstable?

c) For ω > ω_c what is the stable equilibrium value of θ?

d) What is the frequency Ω of small oscillations about this point?

SOLUTION

We may compute the Lagrangian by picking two appropriate orthogonal coordinates θ and φ, where  equals a constant (see Figure 1.2).

where we consider U_eff = -(1/2) ml²ω² sin²θ - mgl cos θ.

=Figure 1.2

a) Employing the usual Lagrange equations

we have

(1)

b) (1) has stationary points where ∂U_eff/∂θ = 0.

To check these points for stability, take the second derivative

(2)

At θ = 0 (2) becomes

So at angular velocities  the potential energy has a minimum and the θ = 0 equilibrium point is stable. However at this point is no longer stable. At θ = π:

This point is unstable for all values of ω.

c) At  (2) becomes

So, here at  the equilibrium point is stable.

d) Consider the initial differential equation (1) and substitute for θ, θ → θ₃+ δ, where θ₃ = cos^-1(g/ω²):

For small oscillations, we will use the approximations sin δ = δ, cos δ = 1-(1/2)δ² and leave only terms linear in δ:

After substituting cos θ₃ = g/lω², we will have for the frequency Ω of small oscillations about this point