Problem $3[10$ Marks$]$

$($a$)$ A $25$ mm $(02-$series$)$ deep$-$groove ball bearing carries a combined load of $2$ kN radially and $3$ kN axially at $1500$ rpm $.$ The outer ring rotates, and the load is steady. Determine the rating life in hours. $[5$ Marks$].$

$($b$)$ A shaft with two supports is loaded in such a way that the axial thrust on each bearing is $500$ N and the radial thrust is $2000$ N and $1000$ N

${}_{F_r}=2000N^{F_a}=500N{}_{F_r}=1000N^{F_a}=500N$

respectively at left$-$ and right$-$hand bearing. The shaft rotates at $1440$ rpm $.$ The bearing is to be selected based on $3$ hours per day for $10$ years. The shaft diameter is $40$ mm but may be reduced if the situation permits considering low moment. Select the suitable bearings. $[5$ Marks$].$

Problem $4.[10$ Marks$]$

$($a$)$ A solid shaft of diameter $D=50mm$ rotates and carries the minor shock loading as shown in figure. Calculate the fatigue safety factor n and yielding factor of safety $n_y$ for the shaft. Given: $D=75mm,S_y=550$MPa, $S_u=660$MPa. The torque fluctuates $5\%$ each way from the mean value, and the shaft is to be machined from unnotched steel. $[5$ Marks$]$

$($b$)$ Determine the value of the critical speed of rotation for the simply supported shaft carrying two loads at C and D as illustrated in figure. $D=50mm,E=210$GPa and density $=7800k\frac{g}{m^3}.[5$ Marks$]$