A lady wears an ear-ring suspended by a light wire of length 3 cm ....

The lady is in a rotating frame of reference, so the ear-ring experiences a centrifugal acceleration in addition to gravity.

Centripetal acceleration of the merry-go-round is given by:

$a = \dfrac{v^2}{R} = \dfrac{4^2}{2} = 8 \text{ m/s}^2$

The effective acceleration due to gravity $g_{\text{eff}}$ is the vector sum of the acceleration due to gravity $g$ and the horizontal acceleration $a$:

$g_{\text{eff}} = \sqrt{g^2 + a^2}$

Taking $g = 10 \text{ m/s}^2$:

$g_{\text{eff}} = \sqrt{10^2 + 8^2} = \sqrt{164} \approx 12.8 \text{ m/s}^2$

The time period of small oscillations for a simple pendulum is:

$T = 2\pi \sqrt{\dfrac{l}{g_{\text{eff}}}}$

Given $l = 3 \text{ cm} = 0.03 \text{ m}$:

$T = 2\pi \sqrt{\dfrac{0.03}{12.8}} \approx 2 \times 3.14 \times 0.0484 \approx 0.30 \text{ s}$