Forces on an Incline

Free-body diagram, friction and Newton's second law

Setup

Angle θ

Mass m

Static friction μ_s

Kinetic friction μ_k

Applied push P (+ = up the slope)

Show

Components of the weight Net force

Result

Weight mg — along the slope, mg sin θ — into the slope, mg cos θ Normal force N Friction f Applied push P Net force (along the slope) Acceleration a Speed0 m/s

Newton's second law along the slope

What's happening

Key idea

Tilt your axes to match the slope. Perpendicular to it, nothing accelerates, so N = mg cos θ. Along it, ΣF = ma. Static friction is only as big as it needs to be, up to μ_sN. Once the block slides, friction is exactly μ_kN.