Servo motor length and diameter reason

A servo motor is used in control systems that need a fast response to the error signal, such as starting, stopping and reversing in a very short time. This requires a high torque-to-inertia ratio, i.e. a small mechanical time constant.

1. Torque produced is proportional to the volume of the rotor: $T \propto D^2 L$ (D = diameter, L = length). So the required torque can be obtained with a long rotor instead of a wide one.
2. Moment of inertia of a rotor is $J \propto D^4 L$. The inertia depends on the fourth power of the diameter but only linearly on the length.
3. So reducing the diameter lowers the inertia drastically, while the torque is maintained by increasing the length. The torque-to-inertia ratio $\propto \frac{1}{D^2}$, which increases as $D$ is made smaller.
4. A low-inertia rotor accelerates and decelerates quickly, so the motor follows the control signal accurately with little overshoot. The small diameter also means the rotor stores little kinetic energy, which makes quick reversal possible.

Hence the servo motor is built long and thin, unlike normal motors, which are designed for efficiency and economy rather than speed of response.