A simple pendulum is placed inside a lift, which is moving with a uniform acceleration. If the time periods of the pendulum while the lift is moving upwards and downwards are in the ratio 1:2, then the acceleration of the lift is (Acceleration due to gravity, $g=10{\text{ ms}}^{-2}$)

Given,

Time period of the pendulum while the lift is moving upwards and downward are in the ratio.

${\text{T}}_{\text{1}}:{\text{T}}_{\text{2}}=1:2$

Acceleration due to gravity, g = 10${\text{m/s}}^{\text{2}}$

We know that,

If the lift is moving upward, then total time-period.

${\text{T}}_{\text{1}}=\text{2}\pi \sqrt{\frac{l}{\text{g}+\text{a}}}$….. (i)           

When the lift is moving downwards, then the total time period,

${\text{T}}_{\text{2}}=\text{2}\pi \sqrt{\frac{l}{\text{g}-\text{a}}}$         ……. (ii)

By dividing Eq. (i) to (ii), we get 

$\therefore \frac{{\text{T}}_{\text{1}}}{{\text{T}}_{\text{2}}}=\sqrt{\frac{\text{g}-\text{a}}{\text{g}+\text{a}}}$

Square on the both sides, we get

or ${\left(\frac{1}{2}\right)}^2=\frac{\text{g}-\text{a}}{\text{g}+\text{a}}\text{or}\frac{\text{g}-\text{a}}{\text{g}+\text{a}}=\frac{1}{4}$ 

or 4g – 4a = g + a or 3g = 5a

or$\text{a}=\frac{\text{3g}}{\text{5}}\Rightarrow \text{a}=\frac{30}{5}=6{\text{ m/s}}^{\text{2}}$ 

So, the acceleration of the lift is 6 ${\text{m/s}}^{\text{2}}$