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

A stick of length L and mass m is in equilibrium while standing on its end A when the end B is gently nudged to the right, causing the stick to fall. Letting μs be the coefficient of static friction between the stick and the ground and modeling the stick as a uniform slender bar, find the largest value of μs for which the stick slides to the left as well as the corresponding value of θ at which sliding begins. To solve this problem, follow the steps below.

(a) Let F and N be the friction and normal forces, respectively, between the stick and the ground, and let F be positive to the right and N positive upward. 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. When slip is impending (i.e., when |F| = μs |N|), |F/N| must be equal to the static coefficient of friction. Therefore, compute the maximum value of | F/N| by differentiating it with respect to θ and setting the resulting derivative equal to zero.

Figure P8.48