A magnet freely suspended in a vibration magnetometer makes 40...

Time period of magnet, $T=2 \pi \sqrt{\frac{I}{M H}}$ In both the condition magnet is same. So, $$\begin{aligned} T & \propto \frac{1}{\sqrt{H}} \\ \frac{T_A}{T_s} & =\frac{\sqrt{H_n}}{H_A}\end{aligned}$$\begin{gathered}\frac{60 / 40}{60 / 20}=\sqrt{\frac{H_n}{36 \times 10^{-4}}} \\ \frac{1}{2}=\frac{\sqrt{H_n}}{36 \times 10^{-6}}\end{gathered}$Squaring both sides$\frac{1}{4}=\frac{H_n}{36 \times 10^{-6}} \Rightarrow H_n=9 \times 10^{-4} \mathrm{~T}$