A 25 kg box of textbooks rests on a loading ramp that makes an angle α with the horizontal. The coefficient of kinetic friction is 0.25, and the coefficient of static friction is 0.35. As the angle is increased, find the minimum angle at which the box starts to slip. 24.5°

18.5°

21.3°

19.3°

Up until the moment the box starts to slip, the static friction is maximized with magnitude f, so that by Newton's second law,

• the net force acting on the box parallel to the ramp is

∑ F = mg sin(α) - f = 0

where mg sin(α) is the magnitude of the parallel component of the box's weight; and

• the net force acting perpendicular to the ramp is

∑ F = n - mg cos(α) = 0

where n is the magnitude of the normal force and mg cos(α) is the magnitude of the perpendicular component of weight.

From the second equation we have

n = mg cos(α)

and f = µn = µmg cos(α), where µ is the coefficient of static friction. Substituting these into the first equation gives us

mg sin(α) = µmg cos(α)   ==>   µ = tan(α)   ==>   α = arctan(0.35) ≈ 19.3°

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