A simple pendulum has time period $T _ { 1 }.$ The point of suspension is now moved upward according to equation $y = K t ^ { 2 }$ where $K = 2.5 \mathrm { m } / \mathrm { s } ^ { 2 }.$ If new time period is $T _ { 2 }$ then ratio will be

Given,

Displacement,

$y=k{{t}^{2}}$

$\dfrac{{{d}^{2}}y}{d{{t}^{2}}}=2k$

Since, k=1m/s²

$\dfrac{{{d}^{2}}y}{d{{t}^{2}}}={{a}_{y}}=2\,m/{{s}^{2}}$

${{T}_{1}}=2\pi \sqrt{\dfrac{l}{g}}.........(1)$

$and$

${{T}_{2}}=2\pi \sqrt{\dfrac{l}{g+{{a}_{y}}}}.........(2)$

Taking ratios of equation (1) and (2)

$\dfrac{{{T}^{2}_{1}}}{{{T}^{2}_{2}}}=\dfrac{g+{{a}_{y}}}{g}=\dfrac{10+2}{10}=\dfrac{6}{5}$