classical mechanics problem: box on an incline

for some reason i'm in a physics mood so i thought i'd share a classic problem that anyone that took elementary mechanics would come across at some point and how to solve it .

Q: a box of mass m sits motionless on an incline. what is the maximum angle Θ of the incline before the box begins to slide down?

A: all good little physics students know the first step is to draw a diagram and then add forces:


where Θ is the angle of the incline, μ is the coefficient of friction and N is the normal force

now lets disect the horizontal and vertical forces on the box:

ΣFx = horizontal component of gravity - friction = mgsin(Θ) - μN = 0 (box is stationary)

-> mgsin(Θ) = μN

ΣFy = vertical component of gravity - normal force = mgcos(Θ) - N = 0 (box is stationary)

-> mgcos(Θ) = N

we have two equations and three unknowns (Θ, μ, and N). the second equation gives us a N in terms of Θ, so let's plug it into the first equation and then solve for Θ:

mgsin(Θ) = μ(mgcos(Θ))

cancel out mg on both sides and get μ by itself:

μ = sin(Θ)/cos(Θ) = tan(Θ)

bring tan to the other side and we are done:

Θ = arctan(μ)

the solution tells us that the largest angle of incline before the box moves is dependent on the coefficient of friction of the incline.

mercy i loves me some classic mechanics :)