Question #77082

A non-uniform rod of length $L$ (in $\text{m}$) is placed on a smooth horizontal table. One end of the rod is attached to a vertical wire with torsional constant $\kappa$ (in $\text{N}\cdot\text{m}\cdot\text{rad}^{-1}$). The linear mass density of the rod varies as $\lambda(x) = \beta x$, where $x$ is the distance from the attached end and $\beta$ is a positive constant (in $\text{kg}\cdot\text{m}^{-2}$). An angular displacement sensor measures the angle of the rod. The sensor has a constant positive zero error $\theta_0$ (in $\text{rad}$), meaning it reads $\theta_0$ when the wire is untwisted and exerts no torque. The rod is rotated until the sensor reads $\phi_0$ (in $\text{rad}$, with $\phi_0 > \theta_0$) and is released from rest. What is the maximum angular speed $\Omega_{max}$ (in $\text{rad}\cdot\text{s}^{-1}$) of the rod during its subsequent motion?