From hardware to equations
A model keeps the mechanisms that matter for your question and leaves the rest out.
First, picture it
For a motor, voltage does not turn directly into speed. It first changes current; current makes torque; torque accelerates the rotor against friction and load.
What the model says
Electrical inductance and resistance govern current; inertia and damping govern speed. The cart-pole needs position, velocity, pole angle and angular velocity. Those are states: values whose present state plus input determine the immediate future. A nonlinear model can be approximated by a linear model near a stated operating point, but the approximation may fail far away.
L di/dt = v-Ri-K_eω; J dω/dt=K_t i-bω-τ_LMotor voltage v acts through current i before changing shaft speed ω.
Make it concrete
A larger rotor inertia J means the same torque produces less acceleration. Compare how fast the motor can change speed, not just its final speed.
Inspect the lab modelA smooth animation does not prove that a model matches a particular real motor. Parameters and validation matter.
Check your understanding+
Why can an upright cart-pole model fail after a large tilt?
A linearization approximates nonlinear forces only near the upright operating point.