Calculate the gravitational potential energy of the water in a lake of volume 100km^3 (mass=1.0x10^14kg). The lake has an average height of 50m above the hydroelectric generators at the base of the dam holding the water back.
PE=mgh=massabove*g*50 Joules
mgh= 10^14*9.8*50=4.9*10^12
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To calculate the gravitational potential energy of the water in the lake, we will use the formula:
Gravitational Potential Energy (PE) = mass (m) x acceleration due to gravity (g) x height (h)
Given:
Volume of the lake = 100 km^3 = 100 x 10^9 m^3
Mass of the water in the lake = 1.0 x 10^14 kg
Average height of the lake above the hydroelectric generators = 50 m
To find the mass of the water in the lake, we need to convert the volume of the lake to mass. We can do this by using the density of water, which is approximately equal to 1000 kg/m^3.
Mass (m) = Volume (V) x Density (ρ)
m = 100 x 10^9 m^3 x 1000 kg/m^3
m = 1 x 10^14 kg
Now we have all the values we need to calculate the gravitational potential energy.
PE = m x g x h
Where:
m = mass of the water in the lake
g = acceleration due to gravity (approximately 9.8 m/s^2 on Earth)
h = average height of the lake above the hydroelectric generators
Substituting the values:
PE = (1 x 10^14 kg) x (9.8 m/s^2) x (50 m)
PE ≈ 4.9 x 10^16 Joules
Therefore, the gravitational potential energy of the water in the lake is approximately 4.9 x 10^16 Joules.