The computational complexity of computing the forces between bodies in contact is presented. The bodies are restricted to be perfectly rigid bodies that contact at finitely many points. It has been known for some time that under the Coulomb model of friction, some configurations of bodies are inconsistent; that is, no contact forces satisfying the constraints of the Coulomb friction model exist for the configuration. The main result of this paper is a proof that determining if a configuration is inconsistent is an NP-complete problem. An immediate corollary of this proof is that computing the contact forces for a configuration of bodies is NP-hard. Computing contact forces remains NP-hard even if configurations are restricted to be consistent.