Consider a closed circular vortex filament of strength γ and radius a. assume that the center of the circular filament is located at the origin and the filament lies in thex-y plane. using the biot-savart law, obtain an expression for the velocity induced at(0,0,z). note that this corresponds to a general point on the z-axis. what is the velocityinduced at the origin (center of the vortex filament)?

Awall of 0.5m thickness is to be constructed from a material which has average thermal conductivity of 1.4 w/mk. the wall is to be insulated with a material having an average thermal conductivity of 0.35 w/mk so that heat loss per square meter shall not exceed 1450 w. assume inner wall surface temperature of 1200°c and outer surface temperature of the insulation to be 15°c. calculate the thickness of insulation required.

Aheavily insulated piston-cylinder device contains 0.02 m3 of steam at 300 kpa and 200 °c. 1.2 mpa. d this process. team is now compressed in a reversible manner to a pressure of etermine the entropy change and the work done on the steam during this process

Tom is having a problem with his washing machine. he notices that the machine vibrates violently at a frequency of 1500 rpm due to an unknown rotating unbalance. the machine is mounted on 4 springs each having a stiffness of 10 kn/m. tom wishes to add an undamped vibration absorber attached by a spring under the machine the machine working frequency ranges between 800 rpm to 2000 rpm and its total mass while loaded is assumed to be 80 kg a) what should be the mass of the absorber added to the machine so that the natural frequency falls outside the working range? b) after a first trial of an absorber using a mass of 35 kg, the amplitude of the oscillation was found to be 10 cm. what is the value of the rotating unbalance? c) using me-3.5 kg. m, find the optimal absorber (by minimizing its mass). what would be the amplitude of the oscillation of the absorber?