Limiting Equilibrium

Question

A horizontal force F just prevents a 10 kg block from sliding down a rough 37° incline (μ_s = 0.5). Find minimum F (g = 10 m/s²).

RankGuru · Accepted Answer

1. **Objective**: Find $F$ such that static friction ($f_s$) up the plane is at its maximum to prevent sliding.
2. **Forces along Plane** ($+$ is up):
   - Upwards: $F \cos 37^\circ + f_s$
   - Downwards: $mg \sin 37^\circ$
   - Equilibrium: $F \cos 37^\circ + \mu_s N = mg \sin 37^\circ$
3. **Normal Force** ($N$): 
   - $N = mg \cos 37^\circ + F \sin 37^\circ$ (F's vertical component also pushes against the plane).
4. **Substitution**:
   - $0.8 F + 0.5 (100 	imes 0.8 + 0.6 F) = 100 	imes 0.6$
   - $0.8 F + 40 + 0.3 F = 60$
   - $1.1 F = 20$
5. **Solve**: $F = \frac{20}{1.1} \approx 18.18$ N.
6. **Final Result**: The minimum force required is approximately **18.2 N**.

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Limiting Equilibrium

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