Now consider the mechanical system $($rotor$)$:

The sum of the torques acting on the rotor is equal to the rotor inertia, $J,$ times its angular acceleration $($where acceleration is the rate of change of angular velocity$)$:

${\displaystyle \underset{?}{\overset{?}{}}}T=J{}^{}$

The torques acting on the motor are the motor torque resulting from the armature current and the damping torque due to the viscous friction damping $($which opposes the motor torque$)$:

${\displaystyle \underset{?}{\overset{?}{}}}T=Ki-b$

Substitute equation $(7)$ into equation $(8)$

Apply the Laplace Transform to the resulting equation and show that this is equal to:

$I(s)=\frac{(Js+b)(s)}{K}$