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 (Fs). Write down the equation for the static friction force (Fs = 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, Gx, will equal Fs. Similarly the perpendicular G component, Gy, will be equal to N. To find Gx and Gy, 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 Gy and G being ϴ. Inspecting the triangle we find that Gx = Gsin(ϴ) and Gy = G*cos(ϴ). by substituting everything into Fs , we receive: μ = sin(ϴ)/cos(ϴ) = tan(ϴ).
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