A body of weight W on an inclined plane of a being pulled up by a horizontal force P will be on the point of motion up the plane when P is equal to
Consider a body of weight W, resting on a rough inclined plane as shown in Fig.
From FBD of the figure.
ΣFx = 0
P × sin α + W × cos α = N.
ΣFy = 0
W × sin α + μN = P × cos α
W × sin α + μ(P × sin α + W × cos α) = P × cos α
W × sin α + μW × cos α = P × cos α - μP × sin α
\(\displaystyle \begin{array}{l}\therefore P=\dfrac{{W\left( {\sin \alpha +\mu \cos \alpha } \right)}}{{\cos \alpha -\mu \sin \alpha }}\\\\\Rightarrow P=W\left( {\dfrac{{\tan \alpha +\mu }}{{1-\mu \tan \alpha }}} \right)\\\\\Rightarrow P=W\left( {\dfrac{{\tan \alpha +\tan \phi }}{{1-\tan \alpha \tan \phi }}} \right)\;\;\;\;\left[ {\because \mu =\tan \phi } \right]\end{array}\)