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. Letting the coefficient of static friction between the stick and the ground be μs = 0.7 and modeling the stick as a uniform slender bar, find the value of θ at which end A of the stick starts slipping and determine the corresponding direction of slip. As part of the solution, plot the absolute value of the ratio between the friction and normal force as a function of θ .To solve this problem, 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) After substituting the expressions for and into the expressions for F and N, plot |F/N| as a function of θ. For impending slip, |F/N| must be equal to μs. Therefore, the desired value of θ corresponds to the intersection of the plot of |F/N| with the horizontal line intercepting the vertical axis at the value 0.7. After determining the desired value of θ, the direction of slip can be found by determining the sign of F evaluated at the θ computed.

Figure P8.49