Answer
$d = 5.6~cm$
Work Step by Step
We can find the value of $d$:
$T = 2\pi~\sqrt{\frac{I}{mgd}}$
$T = 2\pi~\sqrt{\frac{\frac{1}{12}mL^2+md^2}{mgd}}$
$T = 2\pi~\sqrt{\frac{\frac{1}{12}L^2+d^2}{gd}}$
$T^2 = (4\pi^2)~(\frac{\frac{1}{12}L^2+d^2}{gd})$
$3~T^2~gd = \pi^2~L^2+12~\pi^2~d^2$
$12~\pi^2~d^2-3~T^2~gd+\pi^2~L^2 = 0$
We can use the quadratic formula:
$d = \frac{3T^2g \pm \sqrt{(-3T^2g)^2-(4)(12\pi^2)(\pi^2~L^2)}}{(2)(12\pi^2)}$
$d = \frac{3T^2g \pm \sqrt{9T^4g^2-48\pi^4~L^2)}}{24\pi^2}$
$d = \frac{(3)(2.5)^2(9.8) \pm \sqrt{(9)(2.5)^4(9.8)^2-48\pi^4~(1.0)^2}}{24\pi^2}$
$d = 0.056~m, 1.5~m$
$d = 5.6~cm, 150~cm$
Since $150~cm$ is too large, the correct solution is $~~d = 5.6~cm$
