Question
A block of mass m, lying on a rough horizontal plane, is acted upon by a horizontal force P and simultaneously by another force Q acting at an angle θ from the vertical as shown. The block will remain in equilibrium if the coefficient of friction between the block and the surface S is :
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Solution
[Key Concept: Static friction can supply any force up to μN, so equilibrium needs the required friction not to exceed μ times the normal reaction.]
$$\begin{matrix} & & \Sigma F_{y} = 0 \\ & \text{ }N = mg + Qcos\theta & \text{(i)} \\ & \Sigma{\text{ }F}_{x} = 0 & \text{(i)} \\ & Fs = P + Qsin\theta & \text{(ii)} \\ & \ \because fs \leq (fs)_{\max} & \text{(ii)} \\ & P + Qsin\theta \leq \mu N & \text{(ii)} \\ & P + Qsin\theta \leq \mu(mg + Qcos\theta) & \text{(ii)} \\ & \mu \geq \frac{P + Qsin\theta}{mg + Qcos\theta} & \text{(ii)} \end{matrix}$$
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