Inclines with Friction

When an object slides down an incline at angle $\theta$, kinetic friction opposes its motion. Using a tilted coordinate system with axes along and perpendicular to the incline:

The weight $mg$ has components:

• Parallel (down the slope): $mg\sin\theta$
• Perpendicular (into surface): $mg\cos\theta$

The normal force balances the perpendicular component: $N = mg\cos\theta$

Kinetic friction opposes sliding with magnitude $f_k = \mu_k N = \mu_k mg\cos\theta$ and acts up the incline.