Question

A simple pendulum is suspended from the roof of a trolley which moves, in a horizontal direction with an acceleration $a$, and its time period is given by $T=2\pi \sqrt{\frac{l}{g^{\prime }}}$ which of the following correctly represent the value of $g^{\prime }$.

• A. $g$
• B. $g-a$
• C. $g+a$
• D. $\sqrt{g^2+a^2}$

Answer 1

The correct option is D $\sqrt{g^2+a^2}$

Since, the trolley is moving with an acceleration ${}^{\prime }{a}^{\prime }$, a pseudo force $\left(ma\right)$ will act on the simple pendulum, when observed from frame of trolley


Hence net force on bob is,
$F_{net}=m\sqrt{g^2+a^2}$
For effective value of $g$,
$\Rightarrow m{g}^{\prime }=m\sqrt{g^2+a^2}$
$m{g}^{\prime }$ is the effective weight of bob.
$\therefore g^{\prime }=\sqrt{g^2+a^2}$
$\Rightarrow g_{\text{eff}}=g^{\prime }$ is equal to $\sqrt{g^2+a^2}$, will be used in finding time period of pendulum.
$T=2\pi \sqrt{\frac{l}{g^{\prime }}}$