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If you slowly tilt a book with an eraser resting on it, nothing happens at firstβ€”until suddenly, at one exact angle, the eraser snaps free and races down.
Why doesn't it slide gradually, and how do engineers calculate the exact angle where a parked car on an icy street will begin to slide?

Tilting the Coordinate System

When an object moves along an inclined surface, standard horizontal and vertical axes force you to break both acceleration and normal force into messy components.
Instead, we tilt our axes so the parallel axis (xx) runs along the slope and the perpendicular axis (yy) points normal to the surface.
Because the incline is tilted by angle ΞΈ\theta, geometry dictates that the angle between the vertical gravity vector mgmg and the perpendicular axis is also ΞΈ\theta.
This resolves gravity into a perpendicular clamp force FgβŠ₯=mgcos⁑θF_{g\perp} = mg\cos\theta and a downhill driving force Fgβˆ₯=mgsin⁑θF_{g\parallel} = mg\sin\theta.
Since the block cannot sink through the ramp, the normal force balances the perpendicular component: N=mgcos⁑θN = mg\cos\theta.
Gravity tries to pull the object down the rampβ€”so what force holds it stationary until the tilt gets too steep?

Static vs Kinetic Friction on a Slope

In 1699, French physicist Guillaume Amontons discovered that frictional resistance depends directly on the normal force pressing two surfaces together, independent of contact area.
Static friction (fsf_s) is an adjustable holding force that matches whatever downhill pull is applied, up to a maximum threshold called limiting static friction: fs,max⁑=μsNf_{s,\max} = \mu_s N.