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202311091202 Exercise 23.2.8

A block of weight 30\,\mathrm{N} rests on a smooth plane inclined at an angle of 20^\circ to the horizontal. Find the least horizontal force required to keep it in equilibrium and the reaction of the plane.

Extracted from A. Godman & J. F. Talbert. (1973). Additional Mathematics Pure and Applied in SI Units.

Roughwork.

As always, begin with free-body diagrams. Here we combine the two in one force diagram below.

The equation of motion for block m is

\begin{aligned} m\mathbf{a}_{\parallel} & =\mathbf{F}_{\textrm{net},m} \\ & = \mathrm{}_M\mathbf{N}_m + \mathrm{}_E\mathbf{W}_m + \mathbf{f}_m \\ \end{aligned}

along the plane; and for wedge M

\begin{aligned} M\mathbf{a}_{\perp} & =\mathbf{F}_{\textrm{net},M} \\ & = \mathrm{}_E\mathbf{N}_M + \mathrm{}_m\mathbf{N}_M + \mathrm{}_E\mathbf{W}_M + \mathbf{f}_M \\ \end{aligned}

along the earth E. Intuitively, when block m slides leftward down the slope, wedge M will slide rightward on the earth.

This problem is not to be attempted.