A box is at rest on a slope with an angle ϴ. Find an expression for the static friction coefficient, μ, of the box.

Begin by drawing a diagram with all the vectors that act on the box. This should include the normal vector (N), the weight of the box (G), and the static friction force (F_s). Write down the equation for the static friction force (F_s = μN). Using Newton's Second Law of Motion, the forces acting on a static body will equal zero and thus the component of vector G that's parallel to the slanted plane, G_x, will equal F_s. Similarly the perpendicular G component, G_y, will be equal to N. To find G_x and G_y, a straight triangle is drawn connecting vector G and its components. The triangle is similar to that of the slope in the original diagram with the angle between G_y and G being ϴ. Inspecting the triangle we find that G_x = Gsin(ϴ) and G_y = G*cos(ϴ). by substituting everything into F_s , we receive: μ = sin(ϴ)/cos(ϴ) = tan(ϴ).