Pushing force making an angle 0 to the horizontal is applied on a block of weight W placed on a horizontal table. If the angle of friction is ϕ the magnitude of force required to move the body is equal to

The situation is shown in fig.From figure, tan⁡ϕ=fRor f=Rsin⁡ϕ   …(1)The condition for the block (just) to move is Fcos⁡θ=f=Rtan⁡ϕ   …(2)and Fsin⁡θ+R=Wor R=W−Fsin⁡θ   …(3)From eqs. (2) and (3)Fcos⁡θ=(W−Fsin⁡θ)tan⁡ϕ =Wtan⁡ϕ−Fsin⁡θtan⁡ϕor Fcos⁡θ+Fsin⁡θsin⁡ϕcos⁡ϕ=Wsin⁡ϕcos⁡ϕMultiplying by cos ϕ, we getFcos⁡θcos⁡ϕ+Fsin⁡θsin⁡ϕ=Wsin⁡ϕFcos⁡(θ−ϕ)=Wsin⁡ϕF=Wsin⁡ϕcos⁡(θ−ϕ)