45° incline, friction 0.5: will it slide?

A block starts from rest and slides down an incline of angle 45°; 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 sin45 = 0.7071
   The tendency to slide comes from the component of gravity along the slope (factor sinθ)
2. Friction factor μ·cos45 = 0.3536
   The friction limit comes from the normal force and the friction coefficient (factor μcosθ)
3. a = g(sinθ − μcosθ) = 10×(0.7071 − 0.3536) = 3.5355
   Component exceeds friction → it slides

Why it works

On an incline the downhill pull is mg sinθ and friction can cancel at most μmg cosθ. At 45° the two components are equal, since sin45° = cos45° = √2/2, so each is (√2/2)mg.

Common mistakes

• Thinking that sin45° and cos45° are different, when both equal √2/2.
• Writing a = g(sin45° + μcos45°) and adding friction instead of subtracting it.

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