30° incline, friction 0.2: will it slide?

A block starts from rest and slides down an incline of angle 30°; the coefficient of kinetic friction between the block and the incline is 0.2 (g = 10 m/s²). Find the acceleration of the block and say whether it slides.

Step-by-step

1. Gravity component factor sin30 = 0.5
   The tendency to slide comes from the component of gravity along the slope (factor sinθ)
2. Friction factor μ·cos30 = 0.1732
   The friction limit comes from the normal force and the friction coefficient (factor μcosθ)
3. a = g(sinθ − μcosθ) = 10×(0.5 − 0.1732) = 3.2679
   Component exceeds friction → it slides

Why it works

Gravity pulls the block downhill with mg sinθ while friction can resist with at most μmg cosθ. Here sin30° = 0.5 and cos30° = √3/2, and comparing those two forces decides whether the block moves at all.

Common mistakes

• Using sin30° = √3/2 and cos30° = 0.5, swapping the two values for 30°.
• Taking the normal force as mg instead of mg cos30° = (√3/2)mg.

Related

incline-friction n-phy-incline-1 incline-30-mu-0-5-accel