A hydroelectric dam holds back a lake of surface area 3.0 × 106 m2 that has vertical sides below the water level. The water level in the lake is 150 m above the base of the dam. When the water passes through turbines at the base of the dam, its mechanical energy is converted to electrical energy with 90% efficiency. (a) lf gravitational potential energy is taken to be zero at the base of the dam, how much energy is stored in the top meter of the water in the lake? The density of water is 1000 kg/m3. (b) What volume of water must pass through the dam to produce 1000 kilowatt-hours of electrical energy? What distance does the level of water in the lake fall when this much water passes through the dam?

Solution 79P Data given 6 2 Area A = 3.0 × 10 m Height h = 150 m Efficiency e = 90% Density = 1000 kg /m 3 Step 1: We need yo energy stored at the top of the water Mass of the water is m = A × h × Substituting values we get m = 3.0 × 10 m × 150 m × 1000 kg /m 3 m = 4.5 × 10 kg We know PE = mgh Here the energy at the top the tank is same the mass of the water Hence we have energy as 100% 90% = 10% = 0.10 ‘ 11 E = 4.5 × 10 kg/ 0.10 12 E = 4.4 × 10 J 12 Thus we have energy stored as 4.4 × 10 J Step 2 : We know 1000 KWh = 3.6 × 10 J 9 12 Hence we have distance to be passed for 4.4 × 10 J as d = 3.6 × 10 J/ 4.4 × 10 J 12 d = 8.81 × 10 m4 4 This can be approximated to 9 × 10 m Hence the distance to be passed is 9 × 10 m4