The uniform slender pole rests against a small roller at B. End A will not slip

Question

The uniform slender pole rests against a small roller at B. End A will not slip on the horizontal surface if the coefficient of static friction mu_s is sufficiently large, (a) Determine the required minimum value of mu_s to prevent slipping for any value of theta from theta = 0 to theta = 60 degree and plot mu_s versus theta. From these results find the range of theta for which the pole will be unstable if mu_s = 0.40. (6) At what angle theta is the pole most unstable, and what is the least coefficient of static friction mu_s which would be required to prevent slipping for this angle? The band wrench is useful for loosening and tightening such items as the whole-house water filter E shown. Assume that the teeth of the wrench do not slip on the band at point C and that the band is slack from C to its end D. Determine the minimum coefficient mu of static friction for which the band will not slip relative to the fixed filter.

Explanation / Answer

Let N1 is normal reaction by ground upwards and N2 is normal reaction by rollar towards left

By force balance

N1 =mg

N2=umg

By balancing torque about the lower end of the rod

N2 L/2 = mgL tan theta / 4

umgL/2=mgL cot theta / 4

u =(tan theta)/2

for the condition in part a, theta =60 degree

u =0.5 tan 60 degree = 0.866

if u =0.400,

0.8 = tan theta

Theta = 38 degree

Then it will be unstable for thera greather than 38 degree

b) it will be most unstable for theta =60 degree as length of rod is fixed so theta can not be more than that.

u =0.5 tan theta = 0.866

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