A polymorphic system consists of heterogeneous cores such as CPU, GPU, FPGA, and ASIC cores. A polymorphic thread is compiled for multiple morphisms afforded by these diverse cores. The resulting polymorphic computing system can solve two problems - (1) Polymorphic threads enable more complex, dynamic trade-offs between delay and power consumption. A piecewise cobbling of multiple morphism energy-delay profiles of individual thread morphisms offers a richer energy-delay profile for the entire application. (2) The OS scheduler not only picks a thread to run, it also chooses the thread's morphism. In this work, we propose a scheduler to optimize a class of User Satisfaction Index (USI) functions. We develop a model for a mobile polymorphic embedded system computing platform. We integrate a polymorphic scheduler in this model to assess the application design space offered by polymorphic computing. Several greedy versions of a polymorphic scheduler to improve the user satisfaction driven QoS are explored. We build a polymorphic system simulation platform based on SystemC to validate our theoretical analysis of a polymorphic system. Our polymorphic scheduler is evaluated on a variety of application mix with various metrics. We further discuss the feasibility of USI-based polymorphic scheduler by identifying its strengths and weaknesses in relation to the application design space based on the simulation results.