Question

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 $ .

• A. $2\pi \sqrt{\frac{L}{g c o s \alpha }}$
• B. $2\pi \sqrt{\frac{L}{g s i n \alpha }}$
• C. $2\pi \sqrt{\frac{L}{g}}$
• D. $2\pi \sqrt{\frac{L}{g t a n \alpha }}$

Answer 1

Answer: (A)

See the following force diagram.


Vehicle is moving down the frictionless inclined surface so, its acceleration is $gsin\alpha $ . Since vehicle is accelerating, a pseudo force $m\left(g s i n \alpha \right)$ 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_{n e t}=mg \, \, cos\alpha $ or net acceleration of the bob is $g_{e f f}=gcos\alpha $
$\therefore $ Time period $T=2\pi \sqrt{\frac{l}{g_{e f f}}}=2\pi \sqrt{\frac{1}{g c o s \alpha }}$