A simple pendulum is suspended from the ceiling of an elevator. The elevator is accelerating upwards with acceleration a. The period of this pendulum, in terms of its length L, g and a is:

Question

Quizplus · Accepted Answer

When the elevator is accelerating upwards, an upward pseudo-force F = ma acts on the mass of the pendulum. The net force acting on the pendulum is the vector sum of tension T, weight mg and pseudo force ma, which is given by:
F_net = T - mg + ma
Using Newton’s second law, we get:
F_net = ma = -mL(d²Θ/dt²)

where Θ is the angular displacement of the pendulum from the vertical. So,
-mL(d²Θ/dt²) = T - mg + ma
Rearranging, we get:
(d²Θ/dt²) + (g/L)Θ = a
This is the differential equation of simple harmonic motion with angular frequency ω = sqrt(g/L) and amplitude A = a/ω² = aL/g. The period T of the pendulum is given by:
T = 2π/ω = 2πsqrt(L/g)
Substituting for ω, we get:
T = 2πsqrt(L/(g+ a))
Hence, option B is the correct choice.

A Simple Pendulum Is Suspended from the Ceiling of an Elevator