7. A man is pulling a crate on a frictionless slant. If the weight of the crate follows a normal distribution m ~ N(90, 2²) kg. (1) Determine the distribution of the minimum force T for the man to pull the crate. (2) If the man has a weight of 75kg and the static friction between him and the floor follows another normal distribution µ_s ~ N(0.65,0.01²), which is independent from m, determine the probability that man may fail to pull the crate if he uses the minimum force T.

Solution

(1)

On crate, we have:

Thus, we can obtain:

Ans.

(2)

The maximum friction on A is:

Then, we can have:

The man may fail to pull the crate is T > f or Y > 0, where

Hereby, we can obtain:

Consequently, the probability is:

Ans.