A stick of length L and mass m is in equilibrium while standing on its end A when end B is...

A stick of length L and mass m is in equilibrium while standing on its end A when end B is gently nudged to the right, causing the stick to fall. Model the stick as a uniform slender bar, and assume that there is friction between the stick and the ground. Under these assumptions, there is a value of θ, let’s call it θmax, such that the stick must start slipping before reaching θmax for any value of the coefficient of static friction μs.To find the value of θmax, follow the steps below.

(a) Letting F and N be the friction and normal forces, respectively, between the stick and the ground, draw the FBD of the stick as it falls. Then set the sum of forces in the horizontal and vertical directions equal to the corresponding components of . Express the components of in terms of θ, ,and . Finally, express F and N as functions of θ, ,and .

(b) Use the work-energy principle to find an expression for . Differentiate the expression for with respect to time, and find an expression for .

(c) Substitute the expressions for and into the expressions for F and N to obtain F and N as functions of θ. For impending slip, |F/N| must be equal to the coefficient of static friction. Use this fact to determine θmax.

Figure P8.47