First slide
Angle of repose
Question

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

Moderate
Solution

The situation is shown in fig.

From figure, tanϕ=fRor f=Rsinϕ   …(1)

The condition for the block (just) to move is 

Fcosθ=f=Rtanϕ   …(2)

and Fsinθ+R=W

or R=WFsinθ   …(3)

From eqs. (2) and (3)

Fcosθ=(WFsinθ)tanϕ =WtanϕFsinθtanϕor Fcosθ+Fsinθsinϕcosϕ=Wsinϕcosϕ

Multiplying by cos ϕ, we get

Fcosθcosϕ+Fsinθsinϕ=WsinϕFcos(θϕ)=WsinϕF=Wsinϕcos(θϕ)

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