Mass (say) 1 kilogram. Then we can measure the mass of any other object in the universe by colliding it with the reference object and measuring the accelerations. Additionally, mass relates a body's momentum p to its linear velocity v: and the body's kinetic energy "K" to its velocity: The primary difficulty with Mach's definition of mass is that it fails to take into account the potential energy (or binding energy) needed to bring two masses sufficiently close to one another to perform the measurement of mass. This is most vividly demonstrated by comparing the mass of the proton in the
Angular velocity of bases. For example, changing to the mobile frame: where formula_46 are unit vectors for the frame fixed in the moving body. This example has been made using the Z-X-Z convention for Euler angles. The angular velocity vector formula_47 defined above may be equivalently expressed as an angular velocity tensor, the matrix (or linear mapping) "W" = "W"("t") defined by: This is an infinitesimal rotation matrix. The linear mapping "W" acts as formula_49: A vector formula_51 undergoing uniform circular motion around an axis satisfies: Given the orientation matrix "A"("t") of a frame, whose columns are the moving orthonormal coordinate vectors