A block of mass m is on a rough horizontal surface with coefficient of static friction μ. A force F is applied at an adjustable angle θ above the horizontal. The minimum magnitude of F required to initiate motion is:
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Question

A block of mass m is on a rough horizontal surface with coefficient of static friction μ. A force F is applied at an adjustable angle θ above the horizontal. The minimum magnitude of F required to initiate motion is:

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G1 Physics · Accepted Answer

At impending motion, F cosθ = μN and N = mg - F sinθ. Thus F(cosθ + μ sinθ) = μmg. Therefore F = μmg/(cosθ + μ sinθ). This is minimum when cosθ + μ sinθ is maximum. Its maximum value is √(1+μ²). Hence Fₘᵢₙ = μmg/√(1+μ²).