This paper discusses the scheduling problem in a real-time computing system, such as a wireless sensor node powered by ambient regenerative energy. We consider Rate Monotonic (RM), a well-known preemptive task scheduler with fixed priority assignment. We propose a novel scheduling and power management scheme, called RM-H, adapted to energy harvesting constraints. When feasible, the resulting intermittent computing framework makes it possible to operate in an energy-neutral mode with no energy wastage, no energy starvation, and no missed deadlines. The majority of Real Time Operating Systems (RTOS) employ fixed-priority scheduling, in which programmers designate a suitable static priority level to represent the relative urgency of each task. As a result, by integrating this new scheduler based on RM, RTOSs will be able to develop and meet the increasing demand of energy harvesting technology.