Compliant mechanisms have been studied extensively as an alternative to traditional rigid body design with advantages like part number reduction, compliance, and multistable configurations. Most of the past research on compliant mechanisms has been restricted to the case where they are subject to holonomic constraints. In this paper, we develop a model of a planar compliant mechanism with nonholonomic constraints as a mobile robot that can move on the ground. The only actuation that is assumed is a torque on the system. It is shown that the dynamics of this system is similar to that of a well-known nonholonomic system, called the Chaplygin sleigh, but with an added degree-of-freedom and an additional quartic potential. The interaction of compliance and the nonholonomic constraint lead to multiple stable limit cycle oscillations in a reduced velocity space that correspond to oscillations about different stable physical configurations. These limit cycle oscillations produce motion of the compliant mechanism in the plane with differing characteristics. The modeling framework in this paper can form the basis for the design of underacted mobile compliant nonholonomic robots or mobile robots that incorporate compliant mechanisms as mechanical switches.