Halliday, Resnick & Walker — Equilibrium and Elasticity: Problem Problem_53

In Fig. 12.49, a rectangular slab of slate rests on a bedrock surface inclined at angle \(\theta = 26^{\circ}\). The slab has length \(L = 43\text{ m}\), thickness \(T = 2.5\text{ m}\), and width \(W = 12\text{ m}\), and \(1.0\text{ cm}^3\) of it has a mass of \(3.2\text{ g}\). The coefficient of static friction between slab and bedrock is \(0.39\). (a) Calculate the component of the gravitational force on the slab parallel to the bedrock surface. (b) Calculate the magnitude of the static frictional force on the slab. By comparing (a) and (b), you can see that the slab is in danger of sliding. This is prevented only by chance protrusions of bedrock. (c) To stabilize the slab, bolts are to be driven perpendicular to the bedrock surface (two bolts are shown). If each bolt has a cross-sectional area of \(6.4\text{ cm}^2\) and will snap under a shearing stress of \(3.6 \times 10^8\text{ N/m}^2\), what is the minimum number of bolts needed? Assume that the bolts do not affect the normal force.