A block of mass m is lying on a horizontal surface of coefficient of friction μ. A force F is applied to the block at an angle θ with the horizontal. The block will move with a minimum force F if

Question

A block of mass m is lying on a horizontal surface of coefficient of friction μ. A force F is applied to the block at an angle θ with the horizontal. The block will move with a minimum force F if

Infinity Learn · Accepted Answer

Refer to diagram, Vertical component of F is F $$sin$$ θ and the horizontal component is F $$cos$$ θ. Thus              $$R+F$$ $$sin$$ ⁡θ$$=mg$$  or   $$R=mg$$−F $$sin$$⁡ θFrictional force  μ$$R=$$μ(mg-F$$ $$sin$$ θ$$). Also μ(mg-F$$ $$sin$$ θ$$)=F$$ $$cos$$ θor    $$F=$$μ$$mg($$μ $$sin$$⁡θ$$+$$$$cos$$ ⁡θ$$)         $$(i)F$$ will be minimum if the denominator is maximum, i.e., if ddθ$$($$μ $$sin$$⁡ θ$$+$$$$cos$$ ⁡θ$$)=0or$$ μ$$cos$$ ⁡θ−$$sin$$ ⁡θ$$=0or$$ μ$$=$$$$tan$$ ⁡θ

A block of mass m is lying on a horizontal surface of coefficient of friction $μ$ . A force F is applied to the block at an angle $θ$ with the horizontal. The block will move with a minimum force F if

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