30° incline, friction 0.5: 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 0.5 (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 = 0.433
   The friction limit comes from the normal force and the friction coefficient (factor μcosθ)
3. a = g(sinθ − μcosθ) = 10×(0.5 − 0.433) = 0.6699
   Component exceeds friction → it slides

Why it works

Gravity's downhill component is mg sin30° and friction can resist with at most μmg cos30°. With sin30° = 0.5 and cos30° = √3/2 the two forces are mg/2 and (√3/4)mg, so the block does move.

Common mistakes

• Using sin30° = √3/2 and cos30° = 0.5, swapping the values for 30°.
• Thinking a friction coefficient of 0.5 always stops a block, whatever the incline angle.

Related

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