The period of oscillation of a simple pendulum of length $L$ suspended from the roof of a vehicle which moves without friction down an inclined plane of inclination $\alpha$, is given by
IIT 2000,JEE MAIN 2022, Medium
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(a) See the following force diagram. Vehicle is moving down the frictionless inclined surface so, it's acceleration is $g\sin \theta $. Since vehicle is accelerating, a pseudo force $m(g\sin \theta )$ will act on bob of pendulum which cancel the $\sin \theta $ component of weight of the bob.
Hence net force on the bob is $F_{net} = mg\cos \theta $ or net acceleration of the bob is ${g_{eff}} = g\cos \theta $
$\therefore $ Time period $T = 2\pi \sqrt {\frac{l}{{{g_{eff}}}}} = 2\pi \sqrt {\frac{l}{{g\cos \theta }}} $
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