Question

Each minute, \(73,800 \mathrm{~m}^{3}\) of water passes over a waterfall \(96.3 \mathrm{~m}\) high. Assuming that \(58.0 \%\) of the kinetic energy gained by the water in falling is converted to electrical energy by a hydroelectric generator, calculate the power output of the generator. (The density of water is \(1000 \mathrm{~kg} / \mathrm{m}^{3}\).)

Short answer

The power output of the hydroelectric generator is 679.35 GW.

Step-by-step solution

Calculating the mass of the water

The volume of water passing over the waterfall each minute is \(73,800 \mathrm{~m}^{3}\). Since the density of water is \(1,000 \mathrm{~kg/m}^{3}\), we can use this relationship to calculate the mass of the volume of water. The mass (m) is calculated by the formula \(m = \rho * V\) where \(\rho\) is the density and \(V\) is the volume. Substituting the given values, we get \(m = 1,000 \mathrm{~kg/m}^{3} * 73,800 \mathrm{~m}^{3} = 73,800,000 \mathrm{~kg}\)

Calculating the potential energy of the water

The potential energy (U) of an object of mass \(m\) at a height \(h\) is given by the formula \(U = m*g*h\), where \(g\) is the acceleration due to gravity. Here, the height is \(96.3 \mathrm{~m}\) and \(g\) is approximately \(9.8 \mathrm{~m/s}^{2}\). Substituting these values into the formula, we get \(U = 73,800,000 \mathrm{~kg} * 9.8 \mathrm{~m/s}^{2} * 96.3 \mathrm{~m} = 70,277,640,000,000 \mathrm{~J}\)

Calculating kinetic energy and electrical energy

Assuming all the potential energy is converted into kinetic energy, we can now find out how much of this is converted into electrical energy. It is mentioned that 58.0% of the kinetic energy is converted into electrical energy. This would be \(0.58 * 70,277,640,000,000 \mathrm{~J} = 40,761,031,200,000 \mathrm{~J}\)

Calculating power output

Finally, power output is the conversion of energy per unit time. Here, since we're considering energy conversion per minute, we need to divide the total electrical energy by 60 to convert from joules (J = watt seconds) to watts (power = energy/time). Thus, the power output (P) would be \(P = 40,761,031,200,000 \mathrm{~J} / 60 \mathrm{~s} = 679,350,520,000 \mathrm{~W} or 679.35 \mathrm{~GW}\)

Key concepts

Kinetic Energy

Kinetic energy is the energy of motion. For water cascading over a waterfall, it’s the energy the water gains as it falls. This energy is calculated using the equation \( KE = \frac{1}{2} m v^2 \), where \( m \) is mass and \( v \) is velocity. As water plummets, it accelerates due to gravity, increasing its kinetic energy. In the context of a waterfall, this change in energy often originates from a transformation of potential energy as the water descends.

Potential Energy

Potential energy is the stored energy an object has due to its position. For water at the top of a waterfall, potential energy is linked to its height above the ground. This energy is given by \( PE = mgh \), where \( m \) is mass, \( g \) is the gravitational constant \( (9.8 \, \text{m/s}^2) \), and \( h \) is height. When water falls, this stored energy transforms into kinetic energy, making it a vital part of the hydroelectric process.

Energy Conversion

Energy conversion refers to changing energy from one form to another. In hydroelectric plants, the potential energy of water stored at height is converted into kinetic energy as it falls. Subsequently, this kinetic energy is converted into electrical energy by generators. Efficient energy conversion is crucial, as it determines how much potential energy ends up as useful electrical power. In this problem, 58% of the kinetic energy is converted, illustrating realistic energy transformation percentages.

Power Output

Power output is the rate at which energy is produced or converted. It is measured in watts \((W)\), equating to joules per second. After energy conversion, the power output of a generator is calculated by dividing the converted electrical energy by time. In our exercise, with massive amounts of water turning potential energy into vast electrical energy, the power output is significant at 679.35 GW. Such impressive power levels highlight the efficiency and capacity of hydroelectric systems.