A house is heated in winter by a biogas boiler. The boiler is rated at 70,000 Btu/h. The heating value of biogas is 13,000 Btu/h and the average boiler efficiency is 87 percent. Taking the total win- ter period to be 3000 hours and assuming that the boiler operates at an average load of 75 percent, determine the amount of biogas consumed in therm (1 therm = 100,000 Btu) in the winter season. If the biogas is sold at a unit price of $1.20/therm, determine the cost of this biogas to the homeowner.

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?