Two masses 400 g and 350 g are suspended from the ends of a light string passing over a heavy pulley of radius 2 cm . When released from rest the heavier mass is observed to fall 81 cm in 9 s . The rotational inertia of the pulley is _________ \(kg \cdot m ^2\).\(\left( g =9.8 m / s ^2\right)\)

\(S=u t+\frac{1}{2} a t^2\)
\(a =\frac{2 s}{ t ^2}=\frac{2 \times 0.81}{81}=0.02 m / s ^2\)
\(m _1 g- T _1= m _1 a\)
\(T _2- m _2 g= m _2 a\)
\(\left( T _1- T _2\right) R = I \cdot \frac{ a }{ R }\)
\(\therefore a=\frac{\left(m_1-m_2\right) g}{m_1+m_2+\frac{I}{R^2}}\)
\(0.02=\frac{(400-350)\left(10^{-3}\right) g }{(400+350)\left(10^{-3}\right)+\frac{ I }{ R ^2}}\)
\(\frac{ I }{ R ^2}=\frac{50 \times 10^{-3} g}{0.02}-750 \times 10^{-3}=23.75\)
\(I =23.75 \times 4 \times 10^{-4}=9.5 \times 10^{-3} kg- m ^2\)