53° incline, friction 0.25: will it slide?

A block starts from rest and slides down an incline of angle 53°; the coefficient of kinetic friction between the block and the incline is 0.25 (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 = 0.15
   The friction limit comes from the normal force and the friction coefficient (factor μcosθ)
3. a = g(sinθ − μcosθ) = 10×(0.8 − 0.15) = 6.5
   Component exceeds friction → it slides

Why it works

The downhill component of gravity is mg sin53° and friction can hold back at most μmg cos53°. Textbooks take sin53° = 0.8 and cos53° = 0.6, so the net force along the slope is mg(sin53° − μcos53°).

Common mistakes

• Using sin53° = 0.6 and cos53° = 0.8, which are the values for 37°.
• Forgetting that the normal force is mg cos53° = 0.6mg, not mg.

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