A simple, effective electric load leveling technique is to use excess produced electricity to pump water into a hilltop reservoir during off peak hours and then discharge it during peak electrical demand periods. This simple, old style energy storage concept is increasingly being used to have more effective power generation system by addressing the peak power demand. The drawback to this technique is one has to operate it in a hilly or mountainous region with a water supply. Someone has suggested an alternative that can be applied at any location. In this alternative, a reclaimed concrete mass is to be raised using excess generated electrical power. You can assume density of the reclaimed concrete is the same as concrete. This mass will be raised using a pulley-motor combination when there is excess electrical power production. During the peak power demands the mass will be lowered using a pulley-electric generator to produce power. The effect of the friction in the pulley is a tangential force that operates at the outer pulley circumference that interfaces with the cable. This force, F_fric, is proportional to the mass being lifted by the following relationship: F_fric = 0.005 (lifted weight) Note: the force is given as a function of weight (a force in N) not the mass (in kg). There is a similar loss in the cable-winding drum, but the relationship is F_pulley = 0.004 (lifted weight). These frictional losses are ultimately converted to heat into the environment. The mass will be raised or lowered into a pit dug into the earth's surface. The depth of this pit is 100 m and the mass cannot extend above the earth's surface. Assume the device isothermal during these processes.

1. Determine the footprint (the land area needed) of the pit required to store an excess uniform power production of 800 kW over a 12-hour period if the electric motor has an efficiency of 90%. The stored energy is used to meet a peak power demand over a 4-hour period. The power delivered is not uniform over this 4-hour period and follows a function such that power generated = 400 (kW) +b sin (t/a), where t = time measured in hours, a=(4 hour)/ is constant, and b is another constant measured in kW that you should figure out according to given information. The generator efficiency is 90%. NOTE: It is up to you to determine the geometric shape of the concrete mass. The choice of this variable will affect the lift height and the footprint. You must decide this shape. But do not try to optimize it (optimization will be a different part).