Answer
The fraction of the rotational inertia of the disk that the cockroach has is $~~0.2$
Work Step by Step
Let $I_d$ be the rotational inertia of the disk.
Let $I_c$ be the rotational inertia of the cockroach.
We can use conservation of angular momentum to find an expression for $I_c$:
$L_f = L_i$
$(I_d+I_c)~\omega_f = I_d~\omega_i$
$I_c~\omega_f = I_d~\omega_i-I_d~\omega_f$
$I_c~\omega_f = I_d~(\omega_i-\omega_f)$
$I_c = \frac{I_d~(\omega_i-\omega_f)}{\omega_f}$
$I_c = \frac{I_d~(6.0~rad/s-5.0~rad/s)}{5.0~rad/s}$
$I_c = 0.2~I_d$
The fraction of the rotational inertia of the disk that the cockroach has is $~~0.2$
