53° incline, μ = 0.5: does it slide?

A block is released from rest at the top of an incline of angle 53°; 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 sin53 = 0.8
   The tendency to slide comes from the component of gravity along the slope (factor sinθ)
2. Friction factor μ·cos53 = 0.3
   The friction limit comes from the normal force and the friction coefficient (factor μcosθ)
3. a = g(sinθ − μcosθ) = 10×(0.8 − 0.3) = 5
   Component exceeds friction → it slides

Why it works

The downhill component of gravity is mg sinθ and friction can cancel at most μmg cosθ, so the net push along the slope is mg(sinθ − μcosθ).

Common mistakes

• Using sin53° = 0.6 and cos53° = 0.8, which are the values for 37°.
• Quoting an acceleration without first checking that mg sinθ really beats μmg cosθ.

Related

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