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A block of wood rests on a plane inclined to the horizontal at an angle $\theta$. The smallest force acting parallel to the line of the greatest slope which will just move the block down the plane is $F$. If coefficient of friction is $\mu$, then will the weight of the block be $$W = \frac{\mu cos\theta-sin\theta}{F}$$? I've resolved this by $$Wsin\theta + F = \mu Wcos\theta$$

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Your main equation $(mg)sin\theta + F = \mu (mg) cos\theta$ is correct.

On rearranging you get:

$mg=\frac{F}{\mu cos\theta - sin \theta}$.

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