Final answer:
To find the resultant force on a spur gear, one must add the Cartesian vectors representing each force acting on the gear. This involves summing the corresponding components of these vectors to obtain the resultant force vector.
Step-by-step explanation:
The student's question involves determining the resultant force of two forces acting on a spur gear in contact with other gears. This is a problem typically encountered in physics, specifically in the field of mechanics and vector analysis. When dealing with forces represented as vectors, the Cartesian vector form is used to facilitate the calculation of the resultant.
The Cartesian vector for a force takes the form F = ai + bj + ck, where i, j, and k are unit vectors along the x, y, and z-axes respectively, and a, b, and c are the magnitudes of the force components along these axes. To find the resultant force of two forces, one would simply add the corresponding components of the two force vectors.
For example, suppose we have two forces, F1 = 30.0i N acting horizontally and F2 = 40.0j N acting vertically. The resultant vector sum would be Fnet = F1 + F2 = (30.0i + 0j + 0k) + (0i + 40.0j + 0k) = 30.0i + 40.0j N. If the forces are not perpendicular, then one must decompose each force into its components along the chosen coordinate axes and then add these components vectorially.