In a single-channel queuing model, the customer arrival rate is $$12$$ per hour and the serving rate is $$24$$ per hour. The expected time that a customer is in queue is _______ minutes.

A mass of $$2000$$ kg is currently being lowered at a velocity of $$2$$ $$m/s$$ from the drum as shown in the figure. The mass moment of inertia of the drum is $$150$$ kg-m². On applying the brake, the mass is brought to rest in a distance of $$0.5$$ $$m.$$ The energy absorbed by the brake (in $$kJ$$) is __________________.

A machine element $$XY,$$ fixed at end $$X,$$ is subjected to an axial load $$P,$$ transverse load $$F,$$ and a twisting moment $$T$$ at its free end $$Y.$$ The most critical point from the strength point of view is

In a structural member under fatigue loading, the minimum and maximum stresses developed at the critical point are $$50$$ $$MPa$$ and $$150$$ $$MPa,$$ respectively. The endurance, yield, and the ultimate strengths of the material are $$200$$ $$MPa,$$ $$300$$ $$MPa$$ and $$400$$ $$MPa,$$ respectively. The factor of safety using modified Goodman criterion is