The mechanical system shown is a simple model of a rack-pinion mechanism. The rack rolls without slipping horizontally on the pulley, triggering the pulley's rotational motion. Derive the equation of motion for the system using the rack's displacement x as the variable, and then find the natural frequency of the system.

The mechanical system shown is a simple model of a rack-pinion mechanism. The rack rolls without slipping horizontally on the pulley, triggering the pulley's rotational motion. Derive the equation of motion for the system using the rack's displacement x as the variable, and then find the natural frequency of the system.

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The mechanical system shown is a simple model of a rack-pinion mechanism. The rack rolls without slipping horizontally on the pulley, triggering the pulley's rotational motion. Derive the equation of motion for the system using the rack's displacement x as the variable, and then find the natural frequency of the system.

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