A schematic of a clutch-testing machine is shown. The steel shaft rotates at a constant speed v.An axial load is applied to the shaft and is cycled from zero to P. The torque T induced bythe clutch face onto the shaft is given byT 5 f P (D 1 d)4 where D and d are defined in the figure and f is the coefficient of friction of the clutch face. Theshaft is machined with Sy 5 120 kpsi and Sut 5 145 kpsi. The theoretical stress-concentrationfactors for the fillet are 3.0 and 1.8 for the axial and torsional loading, respectively. Assume the load variation P is synchronous with shaft rotation. With Sy 5 0.3, find themaximum allowable load P such that the shaft will survive a minimum of 106 cycles with afactor of safety of 3. Use the modified Goodman criterion. Determine the corresponding factorof safety guarding against yielding.

ENGR 3341 Probability Theory and Statistics Prof. Gelb Week 5 homework solutions Warm-up problems from textbook: Section 3.3 Problem 9: In this problem, we would like to show that the geometric random variable is memoryless. Let X ▯ Geometric(p). Show that P(X > m+ljX > m) = P(X > l); for m;l 2 f1;2;3;:::g We can interpret this in the following way: remember that a geometric random variable can be obtained by tossing a coin repeatedly until observing the ﬁrst heads. If we toss the coin several times and do not observe a heads, from now on it is like we start all over again. In other words, the failed coin tosses do not impact the distribution of waiting time from now on. The reason for this i