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. A \(\left[ {{{\left( {\frac{{{T_M}}}{T}} \right)}^2} - 1} \right]\frac{{Mg}}{A}\;\;\;\;\;\;\)
2. B \(\;\left[ {1 - {{\left( {\frac{{{T_M}}}{T}} \right)}^2}} \right]\frac{A}{{{M_g}}}\)
3. C \(\;\left[ {1 - {{\left( {\frac{T}{{{T_M}}}} \right)}^2}} \right]\frac{A}{{{M_g}}}\)
4. D \(\;\left[ {{{\left( {\frac{{{T_M}}}{T}} \right)}^2} - 1} \right]\frac{A}{{{M_g}}}\)