A bead of mass m slides without friction on a circular loop of radius r. The loop lies in a vertical plane and rotates about a vertical diameter with constant angular velocity ω.

(a) For angular velocity
ω larger than some critical value ω_{c}, the bead can undergo small
oscillations about an equilibrium point θ_{0} ≠ 0. Find ω_{c
}and θ_{0}(ω).

(b) Obtain the equations of motion for the small oscillations about θ_{0} as a function of
ω
and find the period of the oscillations.

Solution:

- Concepts:

Lagrange's equations - Reasoning:

Lagrange's equations are the equations of motion. - Details of the
calculation:

(a) L = T - U.

T = ½m(r^{2}(dθ/dt)^{2}+ r^{2}sin^{2}θ ω^{2}), U = -mgr cosθ.

L = ½m(r^{2}(dθ/dt)^{2}+ r^{2}sin^{2}θ ω^{2}) + mgr cosθ.

Here r and ω are constants.

We have only one generalized coordinate.

d/dt(∂L/∂(dq_{i}/dt)) - ∂L/∂q_{i}= 0.

Lagrange's equation yields

d^{2}θ/dt^{2}= sinθ cosθ ω^{2}- (g/r) sinθ.

d^{2}θ/dt^{2}|_{θequ}= 0 for a stationary point.

The angles θ = 0, θ = π, or cosθ = g/(rω^{2}) define stationary points.

We need g/(rω^{2}) < 1 for the point θ = cos^{-1}(g/(rω^{2}) to exist.

The critical frequency ω_{c}= (g/r)^{½}.

(b) Consider the equilibrium point cosθ_{0}= g/(rω^{2}).

For a stable point we need a restoring force. Let θ = θ_{0}+ δ.

We need d^{2}δ/dt^{2}= -cδ, with c a positive number.

d^{2}δ/dt^{2}= sin(θ_{0}+ δ)cos(θ_{0}+ δ) ω^{2}- (g/r) sin(θ_{0}+ δ)

= d(sinθ cosθ ω^{2}- (g/r) sinθ)/dθ|_{θ0}*δ (Taylor series expansion)

= (cos^{2}θ_{0}ω^{2}- sin^{2}θ_{0}ω^{2}- (g/r) cosθ_{0})δ.

For a stable point we need cos^{2}θ_{0}ω^{2}- sin^{2}θ_{0}ω^{2}- (g/r) cosθ_{0}to be negative.

But cos^{2}θ_{0}ω^{2}- sin^{2}θ_{0}ω^{2}- (g/r)cosθ_{0}= g^{2}/(r^{2}ω^{2}) - ω^{2}< 0 since g/(rω^{2}) < 1 for the point to exist.

This point is always stable if it exists.

The equation of motion for small oscillations about θ_{0}is

d^{2}δ/dt^{2}= -(ω^{2}- g^{2}/(r^{2}ω^{2}))δ.

The frequency of small oscillations about this point is Ω = (ω^{2}- g^{2}/(r^{2}ω^{2}))^{½}.

The period T = 2π/Ω.