A rod of diameter D = 25 mm and thermal conductivity k = 60 W/m ⋅ K protrudes normally from a furnace wall that is at Tw = 200°C and is covered by insulation of thickness Lins = 200 mm. The rod is welded to the furnace wall and is used as a hanger for supporting instrumentation cables. To avoid damaging the cables, the temperature of the rod at its exposed surface, To, must be maintained below a specified operating limit of Tmax = 100°C. The ambient air temperature is T[infinity] = 25°C, and the convection coefficient is h = 15 W/m2 ⋅ K. Derive an expression for the exposed surface temperature as a function of axial location on the rod if it is 700 mm long and the tip adiabatic, total and plot the temperature as a function of length. Repeat your calculations for an insulation layer only 100 mm thick.

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?