Calculate the time period of a simple pendulum of length L suspended from the roof of a vehicle which moves without friction down an inclined plane of inclination angle $\alpha$. 

See the following force diagram.

Vehicle is moving down the frictionless inclined surface so, its acceleration is $g\sin \alpha .$ Since vehicle is accelerating, a pseudo force $m(g\sin \alpha )$ will act on bob of pendulum which cancel the $\sin \alpha$component of weight of the bob.

Hence net force on the bob is $F_{\text{net }}=mg\cos \alpha$ or net acceleration of the bob is $g_{eff}=g\cos \alpha$ 

$\therefore \text{ Time period }$

$\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{l}{{\mathrm{g}}_{\mathrm{efI}}}}=2\mathrm{\pi }\sqrt{\frac{l}{\mathrm{gcos}\mathrm{\alpha }}}$