Question

A pendulum suspended from the ceiling of a train has a time period "T" when the train is at rest. when the train is accelerated with uniform acceleration of 50m/sec2 the period of oscillation will (a) increase (b)remain constant (c) decrease (d)none of these​

Answer 1

Initially the time period of the pendulum is,

$\sf{\longrightarrow T=2\pi\sqrt{\dfrac{l}{g}}}$

After the train gets accelerated by a, (say) horizontally, then the magnitude of the new net acceleration acting on the pendulum is,

$\sf{\longrightarrow a'=\sqrt{a^2+g^2}}$

because the pendulum experiences $\sf{g}$ vertically downwards and pseudo acceleration $\sf{a,}$ due to acceleration of train, horizontally in opposite direction of motion of train. That is, both $\sf{a}$ and $\sf{g}$ are right angled.

This net acceleration is greater than $\sf{g}$ since $\sf{a\neq0.}$

$\sf{\longrightarrow a'>g}$

$\sf{\longrightarrow \dfrac{l}{a'}<\dfrac{l}{g}}$

$\sf{\longrightarrow 2\pi\sqrt{\dfrac{l}{a'}}<2\pi\sqrt{\dfrac{l}{g}}}$

That is,

$\sf{\longrightarrow T'<T}$

where $\sf{T'}$ is new time period.

So the time period decreases. Hence (c) is the answer.