Question

A simple pendulum of length L and mass m is suspended in a car traveling with angular velocity (w) around a circle of radius R. Find the equilibrium angle as a function of w and the frequency of small oscillations around the equilibrium point(s).

Answer 1

(a)

Let, equilibrium angle =

Tension in the string = T

Force on pendulum in horizontal,

Tsin = mw^2R ......(1)

Force in vertical,

Tcos = mg ......(2)

From eq (1) and (2),

tan = w^2*R / g

= tan^-1 ( w^2*R / g)

(b)

Net force on pendulum, F = mw^2R i + mg j

so magnitude of net acceleration, a = sqrt [(w^2*R)² + g²]

Time period, T = 2*pi*sqrt (L / g_eff)

T = 2*pi*sqrt [L / sqrt ((w^2*R)² + g²)]

frequency of small oscillations around the equilibrium point,

f = 1 / T

f = (1 / 2*pi) * sqrt [sqrt ((w^2*R)² + g²) / L]