8. A force P which follows a normal distribution P ~ N(2, 0.025²) lb is applied to the spool as shown below. Determine the probability that the spool will slide if the weight of the spool follows another independent normal distribution W ~ N(1, 0.05²) lb and the static friction between the ground and the spool is μ_s=0.42.

Solution

Then, we can obtain:

F_f ~ N(1.33, 0.0167²) lb.

Also, we could have:

T ~ N(3.849, 0.0347²) lb.

Also, we have:

And:

Thus, we can obtain:

f_max ~ N(1.2283, 0.0222²) lb.

The probability that the spool will slide is that F_f ≥ f_max or Y ≥ 0, where,

Then, we can obtain:

Therefore, the probability that the spool will slide is:

Ans.