A pendulum made of a uniform wire of cross-sectional area \(A\) has time period \(T\). When an additional mass \(M\) is added to its bob, the time period changes to \(T_M\). If the Young’s modulus of the material of the wire is \(Y\) then \(\frac{1}{Y}\) is equal to:
(\(g=\) gravitational acceleration)
1. \( \left[\left(\frac{{T}_{{M}}}{{T}}\right)^2-1\right] \frac{{Mg}}{{A}} \)
2. \(\left[1-\left(\frac{{T}_{{M}}}{{T}}\right)^2\right] \frac{{A}}{{Mg}} \)
3. \(\left[1-\left(\frac{{T}}{{T}_{{M}}}\right)^2\right] \frac{{A}}{{Mg}} \)
4. \(\left[\left(\frac{{T}_{{M}}}{{T}}\right)^2-1\right] \frac{{A}}{{Mg}}\)

For a simple pendulum, a graph is plotted between its kinetic energy (\(KE\)) and potential energy (\(PE\)) against its displacement \(d\). Which one of the following represents these correctly? (graphs are schematic and not drawn to scale)

1.		2.
3.		4.

A simple pendulum oscillating in air has period \(T\). The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is \(\left(\frac{1}{16}\right)^{th}\) of the material of the bob. If the bob is inside a liquid all the time, its period of oscillation in this liquid is:
1. \( 2 T \sqrt{\frac{I}{14}} \)
2. \( 2 T \sqrt{\frac{I}{10}} \)
3. \(4 T \sqrt{\frac{I}{15}} \)
4. \( 4 T \sqrt{\frac{I}{14}} \)
 

A ring is hung on a nail. It can oscillate, without slipping or sliding (i) in its plane with a time period \(T_1\) and, (ii) back and forth in a direction perpendicular to its plane, with a period \(T_2\). The ratio \(\frac{T_1}{T_2}\) will be:
1. \(\frac{2}{\sqrt{3}} \)
2. \(\frac{\sqrt{2}}{3} \)
3. \( \frac{2}{3} \)
4. \(\frac{3}{\sqrt{2}}\)

If the time period of a \(2\) m long simple pendulum is \(2\) s, the acceleration due to gravity at the place where the pendulum is executing simple harmonic motion is:
1. \(\pi^{2}\) m/s²
2. \(2\pi^{2}\) m/s²
3. \(9.8\) m/s²
4. \(16\) m/s²

Given below are two statements: