What is the maximum value the angle can have if the stick is to remain in equilibrium?

Question

One end of a uniform meter stick is placed against a vertical wall . The other end is held by a lightweight cord that makes an angle with the stick. The coefficient of static friction between the end of the meter stick and the wall is 0.400.

a. What is the maximum value the angle can have if the stick is to remain in equilibrium?
b. Let the angle between the cord and the stick is = 16.0. A block of the same weight as the meter stick is suspended from the stick, as shown, at a distance from the wall. What is the minimum value of for which the stick will remain in equilibrium?
c. When = 16.0, how large must the coefficient of static friction be so that the block can be attached 15.0 from the left end of the stick without causing it to slip?

Answer 1

a)
T is the tension in the cord and th is the angle


The normal force at the wall is T*cos(th)

sum torque at the center
mu is the friction coefficient

T*cos(th)*mu/2=T*sin(th)/2
tan(th)=mu
or th=atan(0.4)
th=21.8 degrees

b)

T*cos(16)*.4/2-m*g*x=T*sin(16)/2
sum torque at the wall
m*g/2+m*g*(1-x)=T*sin(16)