A new force definition is derived which permits a more reasonable description of the nature of centripetal or transverse forces. It has separate expressions for the tangential and normal components of force. Concomitant with this new force definition, a particle that moves normal to the accelerating force has an increased inertia as manifested by its acceleration. That particle can be described as having a relativistic normal mass, \(m_n = \gamma ^2 m\), where \(\gamma = (1 - v^2 /c^2 )^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}} \). The corresponding normal force is \(F_{n} = m_{n} \frac{{{d}V_{n} }}{{{d}t}} = \frac{{gamma^{2} {d}p_{n} }}{{{d}t}}\) where the proper velocity \(V = \gamma v\). This new force definition permits us to make a relativistic generalization of Newton's Second Law of Motion. That in turn allows us to find the mass--energy equivalence in new ways. Consider a test mass that is comprised of two particles each of mass, m. Both particles oscillate at constant speed ν, along a common line, about a stationary centre-of-mass. The effective relativistic mass, μ , at the mass centre, is then computed by comparing its dynamical description with that of the individual particles. Numerous ways of making that comparison are possible. All lead to the same result. If the force is applied along the line of motion of the particles, the common expression for relativistic mass results: μt = 2γ m . Conversely, if the force is normal to that line of internal motion, the inertial mass is greater and μn = 2γ3m. Some implications of that anisotropy are discussed.