A fully actuated system can execute any joint trajectory. However, if a system is under-actuated, not all joint trajectories are attainable. The authors have actively pursued novel designs of under-actuated robotic arms which are both controllable and feedback linearizable. These robots can perform point-to-point motions in the state space, but potentially can be designed to work with fewer actuators, hence with lower cost. With this same spirit, the technical note investigates the property of differential flatness for a class of planar under-actuated open-chain robots having a specific inertia distribution, but driven by only one or two actuators. This technical note addresses the following theoretical question: what placement of one or two actuators will make an n-DOF planar robot differentially flat if it is designed so that its center of mass always lies at joint 2?