Applied Mechanics and Graphic Statics
If a set of given forces are such that their free vectors build a closed polygon, then

The resultant force and resultant couple both may not be zero
The resultant force is zero but resultant couple is not zero
The resultant force is zero but resultant couple may not be zero
The resultant force and resultant couple are always zero

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Applied Mechanics and Graphic Statics
A body of weight ‘w’ placed on an inclined plane is acted upon by a force ‘P’ parallel to the plane which causes the body just to move up the plane. If the angle of inclination of the plane is ‘θ’ and angle of friction is ‘φ’, the minimum value of ‘P’, is

w sin (φ - θ)/cos φ
w sin (θ - φ)/cos φ
w cos (θ + φ)/cos φ
w sinθ cos(θ - φ)/sin φ

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