Question

A pendulum in a non-inertial frame ?

Answer 1

Considering the car as a non-inertial reference frame, the pendulum oscilates around the equilibrium position. This occurs with the rope at angle `phi` such that

`phi = arctan(a/g)`

because with the rope at this angle, the momentum of `m vec alpha = m(-a hat i-g hat j)` regarding the pendulum fulcrum is null.

Around this point, the small oscillation period must be

`T=2pi sqrt(l/norm vec alpha)`

Note that without additional acceleration should be `2pi sqrt(l/g)`

so

`T = 2pi sqrt(l/sqrt(g^2+a^2))`